•PRACTICAL   ENGINEERING   DRAWING 

AND 

THIRD   ANGLE   PROJECTION 


FOR  STUDENTS  IN  SCIENTIFIC,  TECHNICAL,  AND  MANUAL  TRAINING  SCHOOLS 

AND  FOR 

ENGINEERING  AND  ARCHITECTURAL  DRAUGHTSMEN,  SHEET  METAL  WORKERS,  ETC. 


Frederick  flecaton  Willson,  C.E.,  H.JVL, 

Professor  of  Descriptiz<e  Geometry^  Stereotonty  and  Technical  Drawing 

in  the 
John  C,  Green  School  of  Science,  Princeton  University. 


York 
THE     MACMILLAN     COMPANY 

LONDON  :    MACMILLAN   &    CO.,    LTD. 
1898 

ALL     RIGHTS     RESERVED 


COPYRIGHT  1896 

BY 

FREDERICK  N.  WILLSON. 


PREFACE 


IN  this  work  the  author  has  aimed  to  present,  concisely  and  with  illustration  that  should  in  especial 
degree   conduce  to  the  interest   of  a  course,  not  only  the  usual  matter  of  first  books  on  mechanical 
drawing   but   also   some    of  the   additional    topics   which   he    regards    as    essential   to    the   education 
of  a   draughtsman,   and    which   could    be   included   without    having   the    book   exceed,    either    in    size    or 
price,  the  leading  works  on  the  same  subject,  previously  in  the  field. 

Having  in  mind  the  needs,  particularly,  of  those  who  have  to  regard  more  the  immediate  prac- 
tical application  of  projections  in  shop  work  than  the  educational  and  disciplinary  value  of  a  course 
in  Descriptive  Geometry,  the  author  presents  only  the  Third  Angle  Method  of  making  working 
drawings,  now  so  generally  employed  in  American  draughting  offices. 

Since  the  pages  were  electrotyped  an  instrument  of  unusual  convenience  and  merit  has  been 
placed  on  the  market,  a  compass  whose  legs  remain  parallel  during  the  process  of  opening,  and 
which  is  therefore  ready  for  use  at  all  angles.  Its  importance  justifies  a  reference  to  it  here,  since 
it  cannot  conveniently  be  incorporated  under  its  proper  heading  in  the  text. 

The  unity  and  continuity  of  the  course  here  offered  is  not  affected  by  the  lack  of  consecutiveness 
in  the  chapters,  its  issue  in  present  shape  having  been  contemplated  at  the  time  of  writing,  although 
it  originally  appeared  in  the  author's  larger  work — Theoretical  and  Practical  Graphics — in  close  correla- 
tion with  a  course  on  the  Descriptive  Geometry  of  Monge  (First  Angle  Method)  and  its  applications 
in  Shadows,  Perspective,  Trihedrals  and  Spherical  Projections. 

F.  N.  W. 


TABLE   OF   CONTENTS 


NOTE-TAKING,  DIMENSIONING,  ETC. 
Technical  Frec-Hand  Sketching  and  Lettering. — 
Note -Taking  from  Measurement. — Dimension- 
ing. —  Conventional  Representations. 

Pages  5  - 10. 

THE  DRAUGHTSMAN'S  EQUIPMENT. 

The  Choice  and  Use  of  Drawing  Instruments  and 
the  Various  Elements  of  the  Draughtsman's 
Equipment.  —  General  remarks  preliminary  to 
instrumental  work. 

Pages  11-20. 

EXERCISES  FOR  PEN  AND  COMPASS. 
Kinds  and  Signification  of  Lines.  —  Designs  for 
Elementary  Practice  with  the  Right  Line  Pen.  — 
Standard  Methods  of  Representing  Materials. — 
Line  Shading.  —  Plane  Problems  of  the  Right 
Line  and  Circle,  including  Rankine's  and 
Kochansky's  approximations.  —  Exercises  for  the 
Compass  and  Bow  -  pen,  including  uniform  and 
tapered  curves.  —  The  Anchor  Ring. —The  Hy- 
perboloid.  —  A  Standard  Rail  Section. 

Pages  21-38. 

ON  HIGHER  PLANE  CURVES  AND  THE  HELIX. 

Regarding  the  Irregular  Curve.  —  The  Helix.  — 
The  Ellipse,  Hyperbola  and  Parabola,  by  various 
methods  of  construction.  —  Homological  Plane 
Curves.  —  Relief  -  Perspective.  —  Link  -  Motion 
Curves.  —  Centroids.  —  The  Cycloid.  —  The 
Companion  to  the  Cycloid. —  The  Curtate  and 
Prolate  Trochoids.  —  Hypo- ,  Epi-,  and  Peri- 
Trochoids.  —  Special  Trochoids,  as  the  Ellipse, 
Straight  Line,  Lima«;on,  Cardioid,  Trisectrix, 


Involute   and   Spiral   of    Archimedes. — Parallel 
Curves.  —  Conchoid.  —  Quadratrix.  —  Cissoid. 

—  Tractrix.  —  Witch     of     Agnesi.  —  Cartesian 
Ovals.  —  Cassian  Ovals.  —  Catenary.  —  Logarith- 
mic   Spiral.  —  Hyperbolic     Spiral.  —  Lituus.  — 
Ionic  Volute. 

Pages  39-78. 
TINTING  AND  SHADING. 

Brush   Tinting,   Flat  and   Graduated.  —  Masonry, 
Tiling,  Wood   Graining,  River -Beds,  elc.,  with 
brush  alone,  or  in  combined  brush  and  line  work. 
Pages  79-87. 

THE  LETTERING  OF  DRAWINGS. 
Free -Hand   Lettering.  —Mechanical    Expedients. 

—  Proportioning     of     Titles.  —  Discussion     of 
Forms.  —  Half-  Block,  Full  Block  and  Railroad 
Types.  —  Borders    and     how     to    draw     them. 
(Alphabets  in  Appendix). 

Pages  88-96. 

BLUE -PRINTING  AND  OTHER  PROCESSES. 

The  Blue -print  Process.  —  Photo-,  and  other  Re- 
productive Graphic  Processes,  including  Wood 
Engraving,  Cerography,  Lithography,  Photo- 
lithography, Chromo- lithography,  Photo -engrav- 
ing, u  Half-  Tones,"  Photo  -  gravure  and  allied 
processes, — How  to  Prepare  Drawings  for  Illustra- 
tion. 

Pages  97-  103. 

THIRD  ANGLE  PROJECTION.  — WORKING 

DRAWINGS. 

Projections  and  Intersections  by  the  Third  Angle 
Method.  —  The  Development  of  Surfaces,  for 


Sheet  Metal  or  Arch  Constructions.  —  Working 
Drawings  of  Bridge  Post  Connection.  —  Struc- 
tural Iron. —  Spur  Gearing  (Approximate  Invo- 
lute Outlines).  —  Helical  Springs,  Rectangular 
and  Circular  Section.  —  Screws  and  Bolts  (U.  S. 
Standard),  and  Table  of  Proportions. 

Pages  131  - 180. 

AXONOMETRIC  (INCLUDING  ISOMETRIC)   PRO- 
JECTION.— ONE  -  PLANE  DESCRIPTIVE 
GEOMETRY. 

Orthographic  Projection  upon  a  Single  Plane.— 
Axonometric  Projection.  — General  Fundamental 
Problem,  inclinations  known  for  two  of  the  three 
axes.  —  Isometric  Projection  vs.  Isometric  Draw- 
ing.—  Shadows  on  Isometric  Drawings. — Tim- 
ber Framings  and  Arch  Voussoirs  in  Isometric 
View. — One-Plane  Descriptive  Geometry. 

Pages  241-247. 

OBLIQUE  PROJECTION. 

Oblique  or  Clinographic  Projection,  Cavalier  Per- 
spective, Cabinet  Projection,  Military  Perspec- 
tive.—  Applications  to  Timber  Framings,  Arch 
Voussoirs  and  Drawing  of  Crystals. 

Pages  248-250. 

APPENDIX. 

Table  of  the  Proportions  of  Washers.  — Working 
Drawings  of  Standard  100  -  Ib.  Rail,  and  of 
Allen  -  Richardson  Slide  Valve.  —  Designs  for 
Variation  of  Problems  in  Chapters  X,  XV  and 
XVI.  —Alphabets. 

Pages  251  -  268. 


FREE-HAND    DRAWING. 


CHAPTER    II. 

ARTISTIC    AND    TECHNICAL    FREE-HAND    DRAWING.  — SKETCHING     FROM     MEASUREMENT.  — FREE- 
HAND   LETTERING.— CONVENTIONAL    REPRESENTATIONS. 

20.     Drawings,   if  classified   as  to   the    method  of  their  production,   are   either  free-hand  or   mechanical; 


Note  Regarding  the  Non-consecutiveness  of  Page  Numbers  In  this   Work. 

As  already  indicated  in  the  preface  to  this  and  each  of  the  other  "  parts  "  or  sections  of  the  author's 
work  entitled  Theoretical  and  Practical  GmjjhicK,  the  pages  are  printed  from  the  plates  of  the  large  work — 
the  most  convenient  arrangement  for  teachers  using  the  latter  with  classes  supplied  with  the  different  parts, 
but  precluding,  necessarily,  the  consccutiveness  of  page  numbers  throughout  any  one  of  the  sections.  This 
docs  not  in  the  slightest  degree  affect  the  unity  of  either  "  part  "  when  employed  as  an  independent  work, 
its  issue  under  separate  cover  having  been  contemplated  and  provided  for  when  the  large  work  was  in 
preparation.  For  the  convenience  of  any,  however,  who  wish  to  assure  themselves  of  the  completeness  of 
either  volume  in  the  series,  this  slip  is  inserted,  containing  the  numbers  of  the  pages  of  the  large  work 
that  each  section  is  intended  to  include.  (See  page  preceding  the  title-page  for  Table  of  Contents  of 
each  section.) 

No.   1.     Pages  o-10;  88-9(5;  Alphabets,  1-151. 

No.  '2.     Pages  1.31-180,  and  Appendix   ((5  pp.). 

No.  3.     Pages  -39-78  and  Appendix. 

No.  4.     Pages  5-10.3;  1-31-180;  241-250;  Appendix,  including  Alphabets. 

No.  5.     Pages  219-240,  and  (after  September,  1899)  Supplement  (8  pp.)  on  Perspective  of  Reflections. 

No.  G.      Pages  ,39-78;  105-250;  Appendix   (8  pp.)  and   Index. 


object.  Yet  to  attain  a  sufficient  degree  of  skill  in  it  for  all  practical  and  commercial  purposes  is 
possible  to  all,  and  among  them  many  who  could  never  hope  to  produce  artistic  results.  It  is  con- 
fined mainly  to  the  making  of  working  sketches,  conventional  representations  and  free-hand  lettering,  and 
the  equipment  therefor  consists  of  a  pencil  of  medium  grade  as  to  hardness;  lettering  pens  — Falcon 
or  Gillott's  303,  with  Miller  Bros.  "Carbon"  pen  No.  4;  either  a  note-book  or  a  sketch-block  or 
pad;  also  the  following  for  sketching  from  measurement:  a  two -foot  pocket -rule;  calipers,  both 
external  and  internal,  for  taking  outside  and  inside  diameters;  a  pair  of  pencil  compasses  for  making 
an  occasional  circle  too  large  to  be  drawn  absolutely  free-hand;  and  a  steel  tape-measure  for  large 
work,  if  one  can  have  assistance  in  taking  notes,  but  otherwise  a  long  rod  graduated  to  eighths. 


TABLE   OF   CONTENTS 


FREE-HAND    DRAWING. 


CHAPTER    II. 


ARTISTIC     AND    TECHNICAL    FREE-HAND    DRAWING.  — SKETCHING     FROM     MEASUREMENT.  — FREE- 
HAND   LETTERING.— CONVENTIONAL    REPRESENTATIONS. 


20.  Drawings,   if  classified   as  to   the   method  of  their  production,   are   either  free-hand   or  mechanical; 
while   as   to    purpose    they    may   be    working   drawings,   so    fully    dimensioned    that    they    can   he   worked 
from  and   what  they   represent  may   be    manufactured;     or  finished    drawings,   illustrative    or    artistic   in 
character    and    therefore    shaded    either    with   pen   or   brush,   and    having   no   hidden   parts   indicated  by 
dotted  lines   as   in   the   preceding   division.      Finished   drawings   also   lack   figured   dimensions. 

Working  drawings  of  parts  or  "  details "  of  a  structure  are  called  detail  drawings;  while  the 
representation  of  a  structure  as  a  whole,  with  all  its  details  in  their  proper  relative  position,  hidden 
parts  indicated  by  dotted  lines,  etc.,  is  termed  a  general  or  assembly  drawing. 

21.  While    mechanical    drawing    is    involved    in    making  the   various   essential   views  —  plans,   eleva- 
tions  and   sections  —  of   all   engineering  and   architectural   constructions,   and   in   solving  the   problems   of 
form  and   relative   position    arising    in   their   design,  yet,  to   the   engineer,  the  ability  to   sketch   effectively 
and   rapidly,   free-hand,   is   of    scarcely    less    importance    than  to   handle   the   drawing   instruments   skill- 
fully ;     while   the   success   of  an   architect   depends   in   still   greater   measure   upon   it. 

We  must  distinguish,  ho\vever,  between  artistic  and  technical  free-hand  work.  The  architect  must 
be  master  of  both;  the  engineer  necessarily  only  of  the  latter. 

To  secure  the  adoption  of  his  designs  the  architect  relies  largely  upon  the  effective  way  in  which 
he  can  finish,  either  with  pen  and  ink  or  in  water -colors,  the  perspectives  of  exterior  and  interior 
views ;  and  such  drawings  are  judged  mainly  from  the  artistic  standpoint.  While  it  is  not  the 
province  of  this  treatise  to  instruct  in  such  work  a  word  of  suggestion  may  properly  be  introduced 
for  the  student  looking  forward  to  architecture  as  a  profession.  He  should  procure  Linfoot's  Picture 
Making  in  Pen  and  Ink,  Miller's  Essentials  of  Perspective  and  Delamotte's  Art  of  Sketching  from  Nature; 
and  with  an  experienced  architect  or  artist,  if  possible,  but  otherwise  by  himself,  master  the  prin- 
ciples and  act  on  the  instructions  of  these  writers. 

22.  Since   the   camera   makes   it,  fortunately,  no   longer   essential   that   a   civil  engineer   should   be   a 
landscape   artist   as   well,   his   free-hand   work   has   become    more   restricted   in   its   scope   and   more   rigid 
in   its   character,   and    like    that    of   the   machine   designer  it   may   properly   be   called   technical,   from   its 
object.      Yet   to   attain   a   sufficient   degree    of    skill    in    it   for  all   practical   and   commercial   purposes    is 
possible   to   all,  and   among  them   many   who   could   never  hope  to   produce   artistic   results.      It   is  con- 
fined  mainly    to    the   making    of    working  sketches,   conventional  representations  and   free-hand  lettering,   and 
the   equipment   therefor   consists   of  a   pencil   of   medium   grade   as   to   hardness;     lettering  pens  —  Falcon 
or   Gillott's   303,   with    Miller   Bros.    "Carbon"   pen    No.   4;     either    a    note-book    or    a    sketch-block   or 
pad;     also    the    following    for    sketching    from    measurement:     a    two -foot    pocket -rule;     calipers,    both 
external   and   internal,  for  taking   outside   and  inside  diameters;    a   pair   of  pencil  compasses  for  making 
an   occasional   circle   too   large   to   be   drawn  absolutely   free-hand;     and    a    steel  tape-measure   for   large 
work,   if  one   can   have   assistance   in   taking   notes,   but   otherwise  a  long   rod   graduated   to   eighths. 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


23.  In  the  evolution  of  a  machine  or  other  engineering  project  the  designer  places  his  ideas  on 
paper  in  the  form  of  rough  and  mainly  free-hand  sketches,  beginning  with  a  general  outline,  or 
"skeleton"  drawing  of  the  whole,  on  as  large  a  scale  as  possible,  then  filling  in  the  details,  separate 
—  and  larger  —  drawings  of  which  are  later  made  to  exact  scale.  While  such  preliminary  sketches  are 
not  drawn  literally  "  to  scale  "  it  is  obviously  desirable  that  something  like  the  relative  proportions 
should  be  preserved  and  that  the  closer  the  approximation  thereto  the  clearer  the  idea  they  will 
give  to  the  draughtsman  or  workman  who  has  to  work  from  them.  A  habit  of  close  observation 
must  therefore  be  cultivated,  of  analysis  of  form  and  of  relative  direction  and  proportion,  by  all 
who  would  succeed  in  draughting,  whether  as  designers  or  merely  as  copyists  of  existing  construc- 
tions. While  the  beginner  belongs  necessarily  in  the  latter  category  he  must  not  forget  that  his  aim 
should  be  to  place  himself  in  the  ranks  of  the  former,  both  by  a  thorough  mastery  of  the  funda- 
mental theory  that  lies  back  of  all  correct  design  and  by  such  training  of  the  hand  as  shall  facilitate 
the  graphic  expression  of  his  ideas.  To  that  end  he  should  improve  every  opportunity  to  put  in 
practice  the  following  instructions  as  to 

SKETCHING    FROM     MEASUREMENT, 

as  each  structure  sketched  and  measured  will  not  only  give  exercise  to  the  hand  but  also  prove  a 
valuable  object  lesson  in  the  proportioning  of  parts  and  the  modes  of  their  assemblage. 

A  free-hand  sketch  may  be  as  good  a  working  drawing  as  the  exactly  scaled  —  and  usually 
hiked  —  drawing  that  is  generally  made  from  it  to  be  sent  to  the  shop. 

While  several  views  are  usually  required,  yet  for  objects  of  not  too  complicated  form,  and  whose 
lines  lie  mainly  in  mutually  perpendicular  directions,  the  method  of  representation  illustrated  by  Fig. 
7,  is  admirably  adapted,*  and  obviates  all  necessity  for  additional  sketches.  It  is  an  oblique  projection 


(Art.  17)  the  theory  of  whose  construction  will  be  found  in  a  subsequent  chapter,  but  with  regard 
to  which  it  is  sufficient  at  this  point  to  say  that  the  right  angles  of  the  front  face  are  seen  in 
their  true  form,  while  the  other  right  angles  are  shown  either  of  30°,  60°,  or  120°;  although  almost 
any  oblique  angle  will  give  the  same  general  effect  and  may  be  adopted.  Lines  parallel  to  each 
other  on  the  object  are  also  parallel  in  the  drawing. 

Draw  first  the  front  face,  whose  angles  are  seen  in  their  true  form ;  then  run  the  oblique  lines 
off  in  the  direction  which  will  give  the  best  view.  (Refer  to  Figs.  42,  44,  45  and  46.) 

24.  While  Fig.  7  gives  almost  the  pictorial  effect  of  a  true  perspective  and  the  object  requires 
no  other  description,  yet  for  complicated  and  irregular  forms  it  gives  place  to  the  plan  -  and  -  eleva - 
tion  mode  of  representation,  the  plan  being  a  top  and  the  elevation  a  front  view  of  the  object.  And 

*The  figures  in  this  chapter  are  photo -reproductions  of  free-hand  work  and  are  intended  not  only  to  illustrate  the  texl 
J>i»t  also  to  set  a  reasonable  standard  for  sketch  -  notes. 


SKETCHING  FROM  M  EAS  UREMENT. 


if  two  views  are  not  enough  for  clearness  as  many  more  should  be  added  as  seem  necessary,  includ- 
ing what  are  called  sections,  which  represent  the  object  as  if  cut  apart  by  a  plane,  separated  and  a 
view  obtained  perpendicular  to  the  cutting  plane,  showing  the  internal  arrangement  and  shape  of  parts. 

In  Fig.  8  we  have  the  same  object  as  in  Fig.  7,  but  represented  by  the  method  just  mentioned. 
The  front  view  (elevation)  is  evidently  the  same  in  both  Figs.  7  and  8,  except  that  in  the  latter  we 
indicate  by  dotted  lines  the  hidden  recess  which  is  in  full  sight  in  Fig.  7. 

The  view  of  the  top  is  placed  at  the  top  in  conformity  to  the  now  quite  general  practice  as  to 
location,  viz.,  grouping  the  various  sketches  about  the  elevation,  so  that  the  view  of  the  left  end  is. 
at  the  left,  of  the  right  at  the  right,  etc. 


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FREE-HAND      SKETCH     OF     TIMBER      FRAMING,      IN      PLAN      AND      ELEVATION) 


In  these  views,  which  fall  under  Art.  19  as  to  theoretical  construction,  entire  surfaces  are  pro- 
jected as  straight  lines,  as  G  B  C  H  in  the  straight  line  H'  C'.  Were  this  a  metallic  surface  and 
"finished"  or  "machined"  to  smoothness,  as  distinguished  from  the  surface  of  a  rough  casting,  that 
fact  would  be  denoted  by  an  "/"  on  the  line  H'  C'  which  represents  the  entire  surface,  the  cross- 
line  of  the  "/"  cutting  the  line  obliquely,  as  shown. 

CENTRE  -  LINES.  —  DIMENSIONING. 

25.  Dimensioning.  In  sketching,  centre-lines  and  all  important  centres  should  be  located  first, 
and  measurements  taken  from  them  or  from  finished  surfaces. 

Feet  and  inches  are  abbreviated  to  "FT.,"  and  "Is."  as  4  FT.  6|  IN.:  also  written  4'  6f",  and 
occasionally  4  FT.  6f".  A  dimension  should  not  be  written  as  an  improper  fraction,  */•"  for  ex- 
ample, but  as  a  mixed  number,  If".  Fractions  should  have  horizontal  dividing  lines. 

Not  only  should  dimensions  of  successive  parts  be  given  but  an  "over -all"  dimension,  which,  it- 
need  hardly  be  said,  should  sustain  the  axiom  regarding  the  whole  and  the  sum  of  its  parts. 

Dimensions  should  read  in  line  with  the  line  they  are  on,  and  either  from  the  bottom  or  the 
right  hand. 

The   arrow   tips   should   touch   the   lines   between   which   a   distance   is   given. 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


Extension  lines  should  be  drawn  and  the  dimension  given  outside  the  drawing  whenever  such 
course  will  add  to  the  clearness.  (See  D'  F',  Fig.  8.) 

An  opening  should  always  be  left  in  the   dimension  line  for  the  figures. 

In  case  of  very  small  dimensions  the  arrow  tips  may  be  located  outside  the  lines,  as  in  Fig.  9, 
and  the  dimension  indicated  by  an  arrow,  as  at  A,  or  inserted  as  at  B  if  there  is  room. 

Should  a  piece  of  uniform  cross-section  (as,  for  example,  a  rail,  angle -iron,  channel  bar,  Phcenix 
column  or  other  form  of  structural  iron)  be  too  long  to  be  represented  in  its  proper  relative  length 
on  the  sketch  it  may  be  broken  as  in  Fig.  9,  and  the  form  of  the  section  (which  in  the  case  sup- 
posed will  be  the  same  as  an  end  vieio)  may  be  inserted  with  its  dimensions,  as  in  the  shaded 
figure.  If  the  kind  of  bar  and  the  number  of  pounds  per  yard  are  known  the  dimensions  can  be 
obtained  by  reference  to  the  handbook  issued  by  the  manufacturers. 

r.  9. 


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wfflfth  — 

FREE-HAND      SKETCH      OF      A      CHANNEL      BAR. 


The  same  dimension  should  not  appear  on  each  view,  but  each  dimension  must  be  given  at  least 
once  on  some  view. 

Notes  on  Rireted  Work,  Pins,  Bolte,  Screws  and  Nuts.  In  riveted  work  the  "  pitch "  of  the  rivets, 
i.  e.,  their  distance  from  centre  to  centre  ("  c.  to  c.")  should  be  noted,  as  also  that  between  centre 
lines  or  rows,  and  of  the  latter  from  main  centre  lines.  Similarly  for  bolts  and  holes.  If  the  latter 
are  located  in  a  circle  note  the  diameter  of  the  circle  containing  their  centres.  Note  that  a  hole 
for  a  rivet  is  usually  about  one -half  the  diameter  of  the  forged  head. 

In  measuring  nuts  take  the  width  between  parallel  sides  ("width  across  the  flats")  and  abbreviate 
for  the  shape,  as  "  sq.,"  "hex.,"  "oct," 

For  a  piece  of  cylindrical  shape  a  frequently  used  symbol  is  the  circle,  as  4"  O  (read  "four 
inches,  round,"  not  wround,)  for  4"  diameter;  but  it  is  even  clearer  to  use  the  abbreviation  of  the 
latter  word,  viz.,  "  diam." 

In  taking  notes  on  bolts  and  screws  the  outside  diameter  is  sufficient  if  they  are  "standard," 
that  is,  proportioned  after  either  the  Sellers  (U.  S.  Standard)  or  Whitworth  (English  Standard)  sys- 
tems, as  the  proportions  of  heads  and  nuts,  number  of  threads  to  the  inch,  etc.,  can  be  obtained 
from  the  tables  in  the  Appendix.  If  not  "  standard "  note  the  number  of  threads  to  the  inch.  Record 
whether  a  screw  is  right-  or  left-handed.  If  right-handed  it  will  advance  if  turned  clock -wise.  The 
shape  of  thread,  whether  triangular  or  square,  would  also  be  noted. 

Notes  on  Gearing.  On  cog,  or  "gear,"  wheels  obtain  the  distance  between  centres  and  the  number 
of  teeth  on  each  wheel.  The  remaining  data  are  then  obtained  by  calculation. 

Bridge  Notes.  In  taking  bridge  notes  there  would  be  required  general  sketches  of  front  and  end 
view;  of  the  flooring  system,  showing  arrangement  of  tracks,  ties,  guard -beam  and  side -walk;  a 
cross  -  section ;  also  detail  drawings  of  the  top  and  foot  of  each  post -connection  in  one  longitudinal 
line  from  one  end  to  the  middle  of  the  structure.  In  case  of  a  double -track  bridge  the  outside 
rows  of  posts  are  alike  but  differ  from  those  of  the  middle  truss. 


CONVENTIONAL    REP  RE  SENT  ATIO  N  S.  — FREE- HAND    LETTERING. 


All  notes  should  be  taken  on  as  large  a  scale  as  possible,  and  so  indexed  that  drawings  of 
parts  may  readily  be  understood  in  their  relation  to  the  whole. 

The  foregoing  hints  might  be  considerably  extended  to  embrace  other  and  special  cases,  but 
experience  will  prove  a  sufficient  teacher  if  the  student  will  act  on  the  suggestions  given,  and  will 
remember  that  to  get  an  excess  of  data  is  to  err  on  the  side  of  safety.  It  need  hardly  be  added 
that  what  has  preceded  is  intended  to  be  merely  a  partial  summary  of  the  instructions  which  would 
be  given  in  the  more  or  less  brief  practice  in  technical  sketching  which,  presumably,  constitutes  a 
part  of  every  course  in  Graphics ;  and  that  unless  the  draughtsman  can  be  under  the  direction  of 
a  teacher  he  will  be  able  to  sketch  much  more  intelligently  after  studying  more  of  the  theory 
involved  in  Mechanical  Drawing  and  given  in  the  later  pages  of  this  work. 

CONVENTIONAL    REPRESENTATIONS. 

26.  Conventional  representations  of  the  natural  leatures  of  the  country  or  of  the  materials  of 
construction  are  so  called  on  the  assumption,  none  too  well  founded,  that  the  engineering  profession 


has  agreed  in  convention  that  they  shall  indicate  that  which  they  also  more  or  less  resemble.  While 
there  is  no  universal  agreement  in  this  matter  there  is  usually  but  little  ambiguity  in  their  use, 
especially  in  those  that  are  drawn  free-hand,  since  in  them  there  can  be  a  nearer  approach  to  the 
natural  appearance.  This  is  well  illustrated  by  Figs.  10  and  11. 


ii- 


In  addition  to  a  rock  section  Fig.  11  (a)  shows  the  method  of  indicating  a  mud  or  sand  bed 
with  small  random  boulders. 

Water  either  in  section  or  as  a  receding  surface  may  be  shown  by  parallel  lines,  the  spaces 
between  them  increasing  gradually. 

Conventional  representations  of  wood,  masonry  and  the  metals  will  be  found  in  Chapter  VI,  after 
hints  on  coloring  have  been  given,  the  foregoing  figures  appearing  at  this  point  merely  to  illustrate, 
in  black  and  white,  one  of  the  important  divisions  of  technical  free-hand  work.  Those,  however,  who 
have  already  had  some  practice  in  drawing  may  undertake  them  either  with  pen  and  ink  or  in 
colors,  in  the  latter  case  observing  the  instructions  of  Arts.  237-241  for  wood,  while  for  the  river 


10  THEORETICAL    AND    PRACTICAL    GRAPHICS. 

sections  they  may  employ  burnt  umber   undertone  for  the    earthy  bed,  pale  blue    or  india  ink   tint  for 
the  rock,  and   prwsian  blue  for  the  water  lines. 

FREE  -  HAND    LETTERING. 

27.  Although  later  on  in  this  work  an  entire  chapter  is  devoted  to  the  subject  of  lettering,  yet 
at  this  point  a  word  should  be  said  regarding  those  forms  of  letters  which  ought  to  be  mastered, 
early  in  a  draughting  course,  as  the  most  serviceable  to  the  practical  worker. 

Fig.  1.2. 

ABCDEFGHIJKLMNOPQRSTUVWXYZ& 

1234567890 

ABCDEFGHIJKLMNOPQRSTUVWXYZdi 

1234567890 

The    first,   known    as    the    Gothic,    is    the    simplest    form    of   letter,   and    is    illustrated    in    both    its 
vertical  and   inclined  (or  Italic)  forms   in   Fig.    12.      It    is    much    used    in    dimensioning,   as    well    as    for 
12  (a.)        titles.      The  lettering  and  numerals  are  Gothic  in  Figs.  7  and  8,  with  the  exception 
of  the  1    and  4,  which,  by   the  addition  of  feet,  are  no  longer  a  pure  form  although 
enhanced   in   appearance. 

In  Fig.  12  (a)    some    modifications  of  the  forms  of  certain  numerals  are  shown; 
also  the  omission  of  the  dividing  line  in  a  mixed  number,  as  is  customary  in  some  offices. 

For  Gothic  letters  and  for  all  others  in  which  there  is  to  be  no  shading  it  is  well  to  use  a  pen 

with  a  blunt  end,  preferably  "ball -pointed,"  but  otherwise  a  medium  stub,  like  Miller  Bros'.  "Carbon" 

No.   4,   which   gives    the    desired  result    when    used    on    a   smooth   surface  and   without   undue   pressure. 

Fig.    13   illustrates  the   Italic   (or  inclined)   form   of   a  letter   which   when   vertical   is   known    as    the 

Roman.      The   Roman  and   Italic  Roman    are    much   used   on   Government  and   other  map   work,   and  in 

s-igr-  ±3- 

ABCDEFGHIJKLMWOPQRS 

22345         T  II  V  W  X  Y  Z         67890 
a~b  c  d  ef  a  Ivi/i  k  lirvno^  %r  s  t~wv 

the  draughting  offices  of  many  prominent  mechanical  engineers.  Regarding  them  the  student  may 
profitably  read  Arts.  260-262.  Make  the  spaces  between  letters  as  nearly  uniform  as  possible,  and 
the  small  letters  usually  about  three -fifths  the  height  of  the  capitals  in  the  same  line. 

For  Roman  and  other  forms  of  letter  requiring  shading  use  a  fine  pen;  Gillott's  No.  303  for 
small  work,  and  a  "  Falcon "  pen  for  larger. 

A  form  of  letter  much  used  in  Europe  and  growing  in  favor  here  is  the  Soennecken  Round 
Writing,  referred  to  more  particularly  in  Art.  265  and  illustrated  by  a  complete  alphabet  in  the 
Appendix.  The  text -book  and  special  pens  required  for  it  can  be  ordered  through  any  dealer  in 
draughtsmen's  supplies. 


THE    DRAUGHTSMAN'S    EQUIPMENT. 


11 


CHAPTER    III. 

DRAWING  INSTRUMENTS  AND   MATERIALS—INSTRUCTIONS   AS  TO  USE.— GENERAL  PRELIMINARIES 

AND   TECHNICALITIES. 


28.  The  draughtsman's  equipment  for  graphical  work  should  be  the  best  con-     Fig-.  ±-3=. 
sistent  with  his  means.      It    is    mistaken    economy    to    buy    inferior   instruments. 

The  best  obtainable  will  be  found  in  the  end  to    have  been  the  cheapest. 

The  set  of  instruments  illustrated  in  the  following  figures  contains  only  those 
which  may  be  considered  absolutely  essential  for  the  beginner. 

THE    DRAWING    PEN. 

The  right  line  pen  (Fig.  14)  is  ordinarily  used  for  drawing  straight  lines, 
with  either  a  rule  or  triangle  to  guide  it;  but  it  is  also  employed  for  the  draw- 
ing of  curves,  when  directed  in  its  motion  by  curves  of  wood  or  hard  rubber. 
For  average  work  a  pen  about  five  inches  long  is  best. 

The  figure  illustrates  the  most  approved  type,  i.  e.,  made  from  a  single  piece 
of  steel.  The  distance  between  its  points,  or  "nibs,"  is  adjustable  by  means  of 
the  screw  H.  An  older  form  of  pen  has  the  outer  blade  connected  with  the 
inner  by  a  hinge.  The  convenience  with  which  such  a  pen  may  be  cleaned  is 
more  than  offset  by  the  certainty  that  it  will  not  do  satisfactory  work  after  the 
joint  has  become  in  the  slightest  degree  loose  and  inaccurate  through  wear. 

29.  If  the    points    wear    unequally    or    become    blunt  the  draughtsman  may 
sharpen  them  readily  himself   upon  a  fine    oil-stone.    The  process  is  as  follows: 

Screw  up  the  blades  till  they  nearly  touch.  Incline  the  pen  at  a  small  angle 
to  the  surface  of  the  stone  and  draw  it  lightly  from  left  to  right  (supposing 

the  initial  position  as  in  Fig.  16).    Before    reaching    the  right         ||  If 


.  is. 


-.  is. 


end  of  the  stone,  begin  turning  the  pen  in  a  plane  perpendic- 
-  ular  to  the  surface,  and  draw  in  the  opposite  direction  at  the 

same  angle.  After  frequent  examination  and  trial,  to  see  that 
the  blades  have  become  equal  in  length  and  similarly  rounded,  the  process  is 
completed  by  lightly  dressing  the  outside  of  each  blade  separately  upon  the 
stone.  No  grinding  should  be  done  on  the  inside  of  the  blade.  Any  "  burr  " 
or  rough  edge  resulting  from  the  operation  may  be  removed  with  fine  emery 
paper.  For  the  best  results,  obtained  in  the  shortest  possible  time,  a  magnifying  glass  should  be 
used.  The  student  should  take  particular  notice  of  the  shape  of  the  pen  Avhen  new,  as  a  standard 
to  be  aimed  at  when  compelled  to  act  on  the  above  suggestions. 

30.  The  pen  may  be  supplied  with  ink  by  means  of  an  ordinary  writing  pen  dipped  in  the  ink 
and  then  passed  between  the  blades;  or  by  using  in  the  same  manner  a  strip  of  Bristol  board 
about  a  quarter  of  an  inch  in  width.  Should  any  fresh  ink  get  on  the  outside  of  the  pen  it  must 


12 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


be   removed;     otherwise  it  will   be  transferred  to   the   edge   of  the   rule   and   thence  to   the   paper,  caus- 
ing  a  blot. 

31.  As    with    the    pencil,   so    with    the    pen,   horizontal    lines    are    to    be    drawn  from    left  to  riyht, 
while  vertical   or  inclined  lines   are   drawn  either  from   or  toward  the  worker,  according  to  the  position 
of  the  guiding   edge  with   respect  to  the  line  to  be  drawn.      If  the  line 

were  m  n,  Fig.  17,  the  motion  would  be  away  from  the  draughtsman, 
i.  e.,  from  n  toward  m;  while  op  would  be  drawn  toward  the  worker, 
being  on  the  right  of  the  triangle. 

32.  To   make    a    sharply    defined,    clean-cut    line — the    only    kind 
allowable — the   pen  should   be   held   lightly    but    firmly,  with    one  blade 
resting  against  the  guiding   edge,   and   with    both    points    resting   equally 
upon  the  paper,  so  that  they  may  wear  at  the  same  rate. 

33.  The   inclination   of   the   pen  to  the   paper  may   best  be   about  70°.      When   properly   held,  the 
pen   will   make  a  line  about    a    fortieth    of   an    inch    from    the    edge    of   the   rule    or    triangle,   leaving 
visible    a    white  line    of   the    paper    of   that    width.      If,   then,   we    wish  to   connect  two  points   by   an 
inked  straight  line,  the  rule  must  be  so  placed  that  its  edge  will  be  from  them  the   distance  indicated. 

It  need   hardly   be  said   that   a   drawing-pen  should   not  be   pushed. 

The  more  frequently  the  draughtsman  will  take  the  trouble  to  clean  out  the  point  of  the  pen 
and  supply  fresh  ink,  the  more  satisfactory  results  will  he  obtain.  When  through  with  the  pen  clean 
it  carefully,  and  lay  it  away  with  the  points  not  in  contact.  Equal  care  should  be  taken  of  all  the 
instruments,  and  for  cleaning  them  nothing  is  superior  to  chamois  skin. 

DIVIDERS. 

34.  The  hair -spring   dividers   (Fig.  15)   are   employed    in   dividing  lines   and   spacing    off   distances, 
and   are  capable   of  the   most   delicate   adjustment   by  means   of  the  screw   G  and   spring   in  one  of  the 
legs.      When  but   one   pair    of    dividers    is    purchased    the    kind    illustrated   should   have  the  preference 
over  plain   dividers,   which   lack   the   spring.      It   will,   however,  be  frequently   found  convenient  to  have 
at  hand   a  pair   of    each.      Should  the  joint  at  F  become  loose   through   wear  it  can   be   tightened   by 
means   of  a  key   having  two  projections   which   fit  into  the  holes   shown   in  the  joint. 

35.  In  spacing  off  distances  the  pressure  exerted  should  be  the  slightest  consistent  with  the  loca- 
tion of  a  point,  the  puncture  to  be  merely  in  the  surface    of   the    paper,  and    the  points  determined 

by   lightly   pencilled   circles   about  them,  thus          Q 0 .     In   laying  off  several  equal  distances 

along   a  line,  all    the    arcs   described   by   one  s>ler- ls-  leg   of  the   dividers   should   be   on   the 

same  side  of  the  line.      Thus,  in   Fig.  19,  with  b  the  first  centre    of   turning,  the  leg  x  describes  the 


arc   R,   then   rests    and    pivots    on    c    while  the  leg  y    describes    the   arc  S;    x  then  traces    arc    T,   etc. 


THE    COMPASSES.— BOW- PENCIL    AND    PEN. 


13 


COMPASS  SET. 
Fig-.  SO.  Fig-  £1.  Fig.  22. 


36.  The  compasses  (Fig.  20)  resemble  the  dividers  in  form  and  may  be  used  to  perform  the  same 
office,  but  are  usually  employed  for  the  drawing  of  circles.  Unlike  the  dividers  one  or  both  of  the 

legs  of  compasses  are  detachable.  Those  illustrated  have  one  perma- 
nent leg,  with  pivot  or  "needle -point''  adjustable  by  means  of  screw  R. 
The  other  leg  is  detachable  by  turning  the  screw  0,  when  the  pen  leg 
L  M  (Fig.  21)  may  be  inserted  for  ink  work ;  or,  where  large  work  is 
involved,  the  lengthening  bar  on  the  right  (Fig.  22)  may  be  first 
attached  at  O,  and  the  pencil  or  pen  leg  then  inserted  at  /.  The 
metallic  point  held  by  screw  S  is  usually  replaced  by  a  hard  lead, 
sharpened  as  indicated  in  Art  54. 

37.  When  in  use,  the  legs  should  be  bent  at  the  joints  P  and  L, 
so  that  they  will  be  perpendicular  to  the  paper  when  the  compasses  are 
held  in  a  vertical  plane.  The  turning  may  be  in  either  direction,  but 
is  usually  "clock-wise;"  and  the  compasses  may  be  slightly  inclined 
toward  the  direction  of  turning.  When  so  used,  and  if  no  undue 
pressure  be  exerted  on  the  pivot  leg,  there  should  be  but  the  slightest 
puncture  at  the  centre,  while  the  pen  points  having  rested  equally  upon 
the  paper  have  sustained  equal  wear,  and  the  resulting  line  has  been 
sharply  defined  on  both  sides.  Obviously  the  legs  must  be  re -adjusted 
as  to  angle,  for  any  material  change  in  the  size  of  the  circles  wanted. 

The  compasses  should  be  held  and  turned  by  the  milled  head 
which  projects  above  the  joint  N. 

Dividers  and  compasses  should  open  and  shut  with  an  absolutely 
uniform  motion,  and  somewhat  stiffly. 


BOW -PENCIL  AND  PEN. 

38.    For  extremely  accurate  work, 

in  diameters  from  one -sixteenth  of  an  inch  to  about 
two  inches,  the  bow -pencil  (Fig.  23)  and  bow -pen  (Fig. 
24)  are  especially  adapted.  The  pencil -bow  has  a 
needle-point,  adjustable  by  means  of  screw  E,  which 
gives  it  a  great  advantage  over  the  fixed  pivot -point 
of  the  bow -pen,  not  alone  in  that  it  permits  of  more 
delicate  adjustment  for  unusually  small  work,  but  also 
because  it  can  be  easily  replaced  by  a  new  one  in 
case  of  damage;  whereas  an  injury  to  the  other  ren- 
ders the  whole  instrument  useless.  For  very  small 
circles  the  needle-point  should  project  very  slightly  beyond  the  pen- 
point;  theoretically,  by  only  the  extremely  small  distance  the  needle- 
point is  expected  to  sink  into  the  paper. 

The  spring  of  either  bow  should  be  strong;  otherwise  an  attempt 
at  a  circle  will  result  in  a  spiral. 

It  will  save  wear  upon  the  threads  of  the  milled  heads  A  and  C 
if  the  draughtsman  will  press  the  legs  of  the  bow  together  with  his 
left  hand  and  run  the  head  up  loosely  on  the  screw  with  his  right. 


14  THEORETICAL    AND    PRACTICAL    GRAPHICS. 

39.  To    the    above    described — which    we    may    call    the    minimum   set    of  instruments — might    be 
advantageously    added   a    pair    of  bow  -  spacers    (small    dividers  shaped  like  Fig.  24) ;    beam  -  compasses, 
for  extra  large  circles;    parallel -rule;    proportional  dividers,  and   an   extra — and   larger — right-line  pen. 

40.  The  remainder  of  the  necessary  equipment  consists  of  paper;    a  drawing-board;    T-rule;    tri- 
angles  or  "  set   squares ; "    scales ;    pencils ;    India  ink ;    water  colors ;    saucers   for  mixing   ink   or  colors ; 
brushes;    water-glass   and  sponge;     irregular   (or   "French")   curves;    india  rubber;    erasing  knife;     pro- 
tractor;   file  for  sharpening  pencils,  or  a  pad  of  fine  emery  or  sand  paper;    thumb-tacks  (or  "drawing- 
pins  " ) ;    horn  centre,  for  making  a  large  number  of  concentric  circles. 

PAPER     AND     TRACING     CLOTH. 

41.  Drawing  paper  may  be  purchased    by  the    sheet    or    roll,  and  either  unmounted  or  mounted, 
i.  e.,  "backed"  by  muslin  or  heavy  card-board.     Smooth  or  "hot-pressed"  paper  is  best  for  drawings 
in    line -work    only;    but    the    rougher    surfaced,   or  "cold -pressed,"   should   always    be   employed   when 
brush-work    in    ink    or    colors  is    involved:    in  the  latter  case,  also,  either  mounted  paper  should    be 
used  or  the  sheets  "  stretched "  by  the  process  described  in  Art.  44. 

42.    The  names  and  sizes  of  sheets   are :  — 

Cap  13  x  17  Elephant  23  x  28 

Demi   15  x  20  Atlas  26  x  34 

Medium   17  X  22  Columbia  23  x  35 

Royal  19  x  24  Double  Elephant  27  x  40 

Super  Royal  19  x  27  Antiquarian  31  x  53 

Imperial  22  x  30 

43.  There  are  many  makes  of  first-class  papers,  but  the  best  known  and  still  probably  the  most 
used  is  Whatman's.      The  draughtsman's  choice  of  paper  must,  however,  be  determined  largely  by  the 
value  of  the  drawing  to  be  made  upon  it,  and  by  the  probable  usage  to  which  it  will  be  subjected. 

Where  several  copies  of  one  drawing  were  desired  it  has  been  a  general  practice  to  make  the 
original,  or  "  construction "  drawing,  with  the  pencil,  on  paper  of  medium  grade,  then  to  lay  over 
it  a  sheet  of  tracing -cloth,  and  copy  upon  it,  in  ink,  the  lines  underneath.  Upon  placing  the  tracing 
cloth  over  a  sheet  of  sensitized  paper,  exposing  both  to  the  light  and  then  immersing  the  sensitive 
paper  in  water,  a  copy  or  print  of  the  drawing  was  found  upon  the  sheet,  in  white  lines  on  a  blue 
ground — the  well-known  blue-print.  The  time  of  the  draughtsman  may,  however,  be  economized,  as 
also  his  purse,  by  making  the  original  drawing  in  ink  upon  Crane's  Bond  paper,  which  combines  in 
a  remarkable  degree  the  qualities  of  transparency  and  toughness.  About  as  clear  blue -prints  can  be 
made  with  it  as  with  tracing  -  cloth,  yet  it  will  stand  severe  usage  in  the  shop  or  the  drafting -room. 

Better  papers  may  yet  be  manufactured  for  such  purposes,  and  the  progressive  draughtsman  will 
be  on  the  alert  to  avail  himself  of  these  as  of  all  genuine  improvements  upon  the  materials  and 
instruments  before  employed. 

44.  To    stretch    paper    tightly    upon    the   board,  lay  the  sheet  right  side  up,*  place  the  long  rule 
with  its  edge  about  one -half  inch  back  from  each  edge  of  the  paper  in  turn,  and  fold  up  against  it 
a  margin  of   that  width.      Then  thoroughly  dampen  the  back   of  the  paper  with  a  full  sponge,  except 
on  the  folded  margins.      Turning  the  paper  again  face  up    gum  the  margins  with  strong  mucilage  or 
glue,  and  quickly  but  firmly  press  opposite  edges  down  simultaneously,  long  sides  first,  exerting  at  the 
same  time  a  slight  outward  pressure  with  the  hands   to    bring    the    paper    down    somewhat    closer   to 


•The  "right  side"  of  a   sheet    is,   presumably,  that   toward  one,  when  — on   holding   it  up   to   the   light— the   manufacturer's 
name,  in  water- mark,  reads  correctly. 


TRACING-CLOTH— DRAWING    BOARD—T-RULE—TRIANGLES.  15 

the  board.  Until  the  gum  "sets,"  so  that  the  paper  adheres  perfectly  where  it  should,  the  latter 
should  not  shrink;  hence  the  necessity  for  so  completely  soaking  it  at  first.  The  sponge  may  be 
applied  to  the  face  of  the  paper  provided  it  is  not  rubbed  over  the  surface,  so  as  to  damage  it. 
The  stretch  should  be  horizontal  when  drying,  and  no  excess  of  water  should  be  left  standing  on 
the  surface;  otherwise  a  water -mark  will  form  at  the  edge  of  each  pool. 

45.  When  tracing  -cloth  is  used  it  must  be  fastened  smoothly,  with  thumb-tacks,  over  the  drawing 
to  be  copied,  and  the  ink  lining  done  upon  the  glazed  side,  any  brush  work  that  may  be  required — 
either  in  ink  or  colors — being  always  done  upon   the    dull    side    of  the  cloth  after  the  outlining  has 
been  completed. 

If  the  glazed  surface  be  first  dusted  with  powdered  pipe -clay,  applied  with  chamois  skin,  it  will 
take  the  ink  much  more  readily. 

When  erasure  is  necessary  use  the  rubber,  after  which  the  surface  may  be  restored  for  further 
pen -work  by  rubbing  it  with  soapstone. 

Tracing -cloth,  like  drawing  paper,  is  most  convenient  to  work  upon  if  perfectly  flat.  When  either 
has  been  purchased  by  the  roll  it  should  therefore  be  cut  in  sheets,  and  laid  away  for  some  time  in 
drawers  to  become  flat  before  needed  for  use. 

DRAWING   BOARD. 

46.  The  drawing  board  should  be  slightly  larger  than   the    paper    for    which    it    is    designed,  and 
of  the  most  thoroughly  seasoned  material,  preferably  some  soft  wood,  as  pine,  to  facilitate  the  use  of 
the  drawing-pins  or  thumb-tacks.      To  prevent   warping    it    should  have  battens    of   hard  wood  dove- 
tailed into  it  across  the  back,  transversely  to  its  length.      The  back   of  the  board  should  be  grooved 
longitudinally  to  a  depth  equal    to  half   the  thickness    of   the   wood,  which  weakens  the  board  trans- 
versely and  to  that  degree  facilitates  the  stiffening  action  of  the  battens. 

For  work  of  moderate  size,  on  stretched  paper,  yet  without  the  use  of  mucilage,  the  "  panel " 
board  is  recommended,  provided  that  both  frame  and  panel  are  made  of  the  best  seasoned  hard  wood. 

It  will  be  found  convenient  for  each  student  in  a  technical  school  to  possess  two  boards,  one 
20"  x  28"  for  paper  of  Super  Royal  size,  which  is  suitable  for  much  of  a  beginner's  work,  and  another 
28"  X  41"  for  Double  Elephant  sheets  (about  twice  Super  Royal  size),  which  are  well  adapted  to  large 
drawings  of  machinery,  bridges,  etc.  A  large  board  may  of  course  be  used  for  small  sheets,  and  the 
expense  of  getting  a  second  board  avoided;  but  it  is  often  a  great  convenience  to  have  a  medium- 
sized  board,  especially  in  case  the  student  desires  to  do  some  work  outside  the  draughting -room. 

THE    T-RULE. 

47.  The  T-rule  should  be  slightly  shorter    than    the    drawing    board.      Its    head  and  blade  must 
have    absolutely    straight    edges,  and    be    so    rigidly    combined    as    to  admit  of  no  lateral  play  of  the 
latter  in  the  former.     The  head  should  also  be  so  fastened  to  the  blade  as  to  be  level  with  the  surface 
of  the  board.      This    permits   the    triangles    to    slide    freely    over    the    head,  a   great  convenience  when 
the  lines  of  the  drawing  run  close  to  the  edge  of  the  paper.     (See  Fig.  32.) 

The  head  of  the  T-rule  should  always  be  used  along  the  left-hand  edge    of  the  drawing   board. 

TRIANGLES. 

48.  Triangles,   or  "set -squares"  as  they  are  also  called,  can    be  obtained  in  various  materials,   as 
hard  rubber,  celluloid,  pear -wood,  mahogany  and  steel;    and  either  solid  (Fig.  25)   or  open  (Fig.  26). 
The  open  triangles  are  preferable,  and  two  are  required,  one    with    acute  angles  of   30°  and  60°,   the 
other  with  45°  angles.      Hard  rubber    has    an    advantage    over  metal  or  wood,  the  latter  being  likely 
to   warp  and  the  former  to  rust,  unless  plated.      Celluloid  is  transparent  and  the  most  cleanly  of  all. 


16 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


The  most  frequently  recurring  problems  involving  the   use  of  the  triangles  are  the  following:  — 


Fig.  2S. 


49.  To  draw  parallel  lines  place  either  of  the  edges 
against  another  triangle  or  the  T-rule.  If  then  moved 
along,  in  either  direction,  each  of  the  other  edges  will 
take  a  series  of  parallel  positions. 

•  50.  To  draw  a  line  perpendicular  to  a  given  line, 
place  the  hypothenuse  of  the  triangle,  o  a,  (Fig.  26), 
so  as  to  coincide  with  or  be  parallel  to  the  given 
line;  then  a  rule  or  another  triangle  against  the  base.  By  then  turning  the  triangle  so  that  the 
other  side,  o  c,  of  its  right  angle  shall  be  against  the  rule,  as  at  olcl,  the  hypothenuse  will  be  found 
perpendicular  to  its  first  position  and  therefore  to  the  given  line. 

51.      To    construct    regular    hexagons    place    the    shortest    side    of    the    60° 
triangle  against  the   rule   (Fig.  27)   if   two    sides  are  to   be   horizontal,   as  fe 

H      ^    I      H      I  and  b  c  of  hexagon   H.      For   vertical  sides,   as   in   H',   the    position    of  the 

triangle  is   evident.      By   making  ab  indefinite  at  first,   and    knowing  be — 
^Lgu     the   length   of  a  side,   we  may   obtain  a  by   an  arc,   centre  b,   radius   b  c. 

3  If  the  inscribed  circles  were  given,  the  hexagons  might  also  be  obtained 

by  drawing  a  series  of  tangents  to  the  circles,  with  the  rule  and  triangles  in  the  positions  indicated. 

THE    SCALE. 

52.  But  rarely  can  a  drawing  be  made  of  the  same  size  as  the  object,  or  "full-size,"  as  it  is 
called;  the  lines  of  the  drawing,  therefore,  usually  bear  a  certain  ratio  to  those  of  the  object.  This 
ratio  is  called  the  scale,  and  should  invariably  be  indicated. 

If  six  inches  on  the  drawing  represent  one  foot  on  the  object,  the  scale  is  one-half  and  might  be 
variously  indicated,  thus:  SCALE  |;  SCALE  1:2;  SCALE  6  IN.  —  1  FT.  SCALE  6"  =  !'. 

At  one  foot  to  the  inch  any  line  of  the  drawing  would  be  one -twelfth  the  actual  size,  and  the 
fact  indicated  in  either  of  the  ways  just  illustrated. 

Although  it  is  a  simple  matter  for  the  draughtsman  to  make  a  scale  for  himself  for  any  par- 
ticular case,  yet  scales  can  be  purchased  in  great  variety,  the  most  serviceable  of  which  for  the  usual 
range  of  work  is  of  box-wood,  12"  long,  (or  18",  if  for  large  work)  of  the  form  illustrated  by  Fig. 
'-  ss-  28,  and  graduated  -fc  :  •&  :  | :  {•  :  | :  \  :  f  :  1  :  \\  :  3  inches  to  the  foot.  This 

is  known   as   the   architect's  scale,  in   contradistinction   to    the   engineer's,   which  is 
decimally    graduated.      It    will,    however,    be    frequently    convenient    to    have    at 
hand  the  latter  as  well  as  the  former. 
When    in    use    it    should    be    laid    along    the  line    to    be   spaced,   and   a  light   dot   made   upon  the 


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SCALES.  — PENCILS.  — INKS.  17 

be  transferred  from  the  scale  to  the  drawing    by  the  dividers,  as    such   procedure    damages    the    scale 
if  not  the  paper. 

53.  For    special    cases    diagonal    scales    can    readily    be    constructed.      If,    for    example,   a    scale    of 
3  inches  to   the   foot  is   needed   and   measuring   to  fortieths    of  inches,   draw    eleven    equidistant,   parallel 
lines,  enclosing   ten    equal    spaces,    as    in    Fig.  29,  and    from    the    end  A  lay  off  A  B,  B  C,  etc.,   each 
3  inches    and    representing    a    foot.      Then    twelve    parallel    diagonal    lines    in    the  first  space    intercept 
quarter -inch    spaces    on   AB  or   a  b,   each    representative    of   an    inch.      There    being    ten    equal    spaces 
between   B  and  b,  the   distance  s  x,  of  the  diagonal   b  m  from  the   vertical   b  B,  taken  on  any  horizontal 
line  s  x,  is  as  many  tenths  of  the  space  m  B  as  there  are  spaces  between  s  x  and  b ;    six,  in  this  case. 
The   principle   of  construction  may  be   generalized   as   follows:  — 

The  distance  apart  of  the  vertical  lines  represents  the  units  of  the  scale,  whether  inches,  feet, 
rods  or  miles.  Except  for  decimal  graduation  divide  the  left-hand  space  at  top  and  bottom  into  as 
many  spaces  as  there  are  units  of  the  next  lower  denomination  in  one  of  the  original  units  (feet, 
for  yards  as  units;  inches  in  case  of  feet,  etc.).  Join  the  points  of  division  by  diagonal  lines;  and, 
if  —  is  the  smallest  fraction  that  the  scale  is  designed  to  give,  rule  x  +  1  equidistant  horizontal  lines, 
giving  x  equal  horizontal  spaces.  The  scale  will  then  read  to  jth  of  the  intermediate  denomination 
of  the  scale. 

When  a  scale  is  properly  used,  the  spaces  on  it  which  represent  feet  and  inches  are  treated  as 
if  they  were  such  in  fact.  On  a  scale  of  one -eighth  actual  size  the  edge  graduated  1£  inches  to  the 
foot  would  be  employed;  each  1%  inch  space  on  the  scale  would  be  read  as  if  it  were  a  foot;  and 
ten  inches,  for  example,  would  be  ten  of  the  eighth -inch  spaces,  each  of  which  is  to  represent  an 
inch  of  the  original  line  being  scaled.  The  usual  error  of  beginners  would  be  to  divide  each  original 
dimension  by  eight  and  lay  off  the  result,  actual  size.  The  former  method  is  the  more  expedition 

THE     PENCILS. 

54.  For  construction  lines  afterward  to  be  inked  the  pencils  should    be    of   hard   lead,  grade 

if  Fabers  or  VVH  if  Dixon's.  The  pencilling  should  be  light.  It  is  easy  to  make  a  groove  in  the 
paper  by  exerting  too  great  pressure  when  using  a  hard  lead.  The  hexagonal  form  of  pencil  is 
usually  indicative  of  the  finest  quality,  and  has  an  advantage  over  the  cylindrical  in  not  rolling  off 
when  on  a  board  that  is  slightly  inclined. 

Somewhat  softer  pencils  should  be  used  for  drawings  afterward  to  be  traced,  and  for  the  prelim- 
inary free-hand  sketches  from  which  exact  drawings  are  to  be  made;  also  in  free-hand  lettering. 

Sharpen  to  a  chisel  edge  for  work  along  the  edges  of  the  T-rule  or  triangles,  but  use  another 
pencil  with  coned  point  for  marking  off  distances  with  a  scale,  locating  centres  and  other  isolated 
points,  and  for  free-hand  lettering;  also  sharpen  the  compass  leads  to  a  point.  Use  the  knife  for 
cutting  the  wood  of  the  pencil,  beginning  at  least  an  inch  from  the  end.  Leave  the  lead  exposed 
for  a  quarter  of  an  inch  and  shape  it  as  desired,  either  with  a  knive  or  on  a  fine  file,  or  a  pad  of 
emery  paper. 

THE    INK. 

55.  Although  for  many  purposes    some    of   the    liquid    drawing -inks    now    in    the    market,  partic- 
ularly Higgins',  answer  admirably,  yet  for  the  best   results,  either  with  pen  or  brush,  the   draughtsman 
should    mix    the    ink    himself    with   a  stick  of    India — or,   more   correctly,    China   ink,   selecting   one    of 
the  higher -priced  cakes,  of  rectangular  cross  -  section.      The  best  will  show  a  lustrous,  almost  iridescent 
fracture,  and   will  have  a  smooth,   as   contrasted   with   a  gritty  feel  when   tested  by   rubbing  the  moist- 
ened  finger   on  the   end   of  the   cake. 


18  THEORETICAL    AND    PRACTICAL     GRAPHICS. 

4 

Sets  of  saucers,  called  "nests,"  designed  for  the  mixing  of  ink  and  colors,  form  an  essential  part 
of  an  equipment.  There  are  usually  six  in  a  set,  and  so  made  that  each  answers  as  a  cover  for  the 
one  below  it.  Placing  from  fifteen  to  twenty  drops  of  water  in  one  of  these,  the  stick  of  ink  should 
be  rubbed  on  the  saucer  with  moderate  pressure. 

To  properly  mix  ink  requires  great  patience,  as  with  too  great  pressure  a  mixture  results  having 
flakes  and  sand -like  particles  of  ink  in  it,  whereas  an  absolutely  smooth  and  rather  thick,  slow- 
flowing  liquid  is  wanted,  whose  surface  will  reflect  the  face  like  a  mirror.  The  final  test  as  to 
sufficiency  of  grinding  is  to  draw  a  broad  line  and  let  it  dry.  It  should  then  be  a  rich  jet  black, 
with  a  slight  lustre.  The  end  of  the  cake  must  be  carefully  dried  on  removing  it  from  the  saucer, 
to  prevent  its  flaking,  which  it  will  otherwise  invariably  do. 

One  may  say,  almost  without  qualification,  and  particularly  when  for  use  on  tracing -cloth,  the 
thicker  the  ink  the  better;  but  if  it  should  require  thinning,  on  saving  it  from  one  day  to  another  — 
which  is  possible  with  the  close-fitting  saucers  described — add  a  few  drops  of  water,  or  of  ox -gall  if 
for  use  on  a  glazed  surface. 

When  the  ink  has  once  dried  on  the  saucer  no  attempt  should  be  made  to  work  it  up  again 
into  solution.  Clean  the  saucer  and  start  anew. 

WATER     COLORS. 

56.  The    ordinary    colored    writing    inks    should    never   be   used  by  the  draughtsman.     They  lack 
the   requisite   "  body "   and   are   corrosive  to  the   pen.      Very  good   colored  drawing   inks  are  now  manu- 
factured   for   line  work,  but  Winsor  and  Newton's  water    colors,  in    the    form    called    "moist,"  and  in 
"half -pans,"   are   the   best    if   not    the    most    convenient,  for    color    work    either    with    pen    or    brush. 
Those  most   frequently   employed   in   engineering   and   architectural   drawing  are   Prussian  Blue,  Carmine, 
Light  Red,  Burnt  Sienna,   Burnt  Umber,  Vermilion,  Gamboge,  Yellow  Ochre,    Chrome  Yellow,   Payne's 
Gray  and  Sepia.      For  some  of  their  special  uses  see  Art.  73. 

Although  hardly  properly  called  a  color,  Chinese  White  may  be  mentioned  at  this  point  as  a 
requisite,  and  obtainable  of  the  same  form  and  make  as  the  colors  above. 

DRAWING-PINS. 

57.  Drawing-pins  or  thumb-tacks,  for  fastening  paper  upon  the  board,  are  of  various  grades,  the 
best,  and  at  present  the  cheapest,  being  made  from  a  single  disc  of  metal  one -half  inch  in  diameter, 
from  which  a  section  is  partially   cut,  then  bent  at  right  angles  to  the  surface,  forming  the  point  of 
the  pin. 

IRREGULAR     CURVES. 

58.  Irregular    or    French    curves,   also    called    sweeps,  for    drawing    non- circular    arcs,    are    of  great 
variety,  and    the    draughtsman    can    hardly    have    too    many    of  them.      They    may    be  either  of  pear 

r-  so.  wood   or  hard  rubber.      A  thoroughly  equipped   draughting  office  will  have  a 

large  stock  of  these  curves,  which  may  be  obtained  in  sets,  and  are  known 
as  railroad  curves,  ship  curves,  spirals,  ellipses,  hyperbolas,  parabolas  and 
combination  curves.  Some  very  serviceable  flexible  curves  are  also  in  the 
market. 

If  but  two  are  obtained  (which  would  be  a  minimum  stock  for  a 
beginner)  the  forms  shown  in  Fig.  30  will  probably  prove  as  serviceable  as  any.  When  employing 
them  for  inked  work  the  pen  should  be  so  turned,  as  it  advances,  that  its  blades  will  maintain  the 
same  relation  (parallelism)  to  the  edge  of  the  guiding  curve  as  they  ordinarily  do  to  the  edge  of 


CURVES.  — RUBBER.  — ERASERS.  — PROTRACTORS.  — BRUSHES. 


19 


the  rule.  And  the  student  must  content  himself  with  drawing  slightly  less  of  the  curve  than  might 
apparently  be  made  with  one  setting  of  the  sweep,  such  course  being  safer  in  order  to  avoid  too  close 
an  approximation  to  angles  in  what  should  be  a  smooth  curve.  For  the  same  reason,  when  placed 
in  a  new  position,  a  portion  of  the  irregular  curve  must  coincide  with  a  part  of  that  last  inked. 

The  pencilled  curve  is  usually  drawn  free-hand,  after  a  number  of  the  points  through  which  it 
should  pass  have  been  definitely  located.  In  sketching  a  curve  free-hand  Ft  is  much  more  naturally 
and  smoothly  done  if  the  hand  is  always  kept  on  the  concave  side  of  the  curve. 

INDIA     RUBBER. 

59.  For    erasing    pencil -lines    and    cleaning    the    paper    india    rubber    is    required,   that    known    as 
"  velvet "   being    recommended    for    the    former    purpose,   and    either   "  natural "   or   "  sponge "   rubber   for 
the   latter.      Stale  bread   crumbs   are   equally   good  for  cleaning  the  surface  of  the   paper  after  the  lines 
have  been  inked,   but   will   damage   pencilling  to   some   extent. 

One  end  of  the  velvet  rubber  may  well  be  wedge-shaped  in  order  to  erase  lines  without  damag- 
ing others  near  them. 

INK    ERASER. 

60.  The  double-edged  erasing  knife  gives  the  quickest  and  best  results  when  an  inked  line  is  to 
be  removed.      The    point    should  rarely  be  employed.      The  use    of   the    knife  will  damage  the  paper 
more    or    less,    to    partially    obviate    which    rub    the    surface    with    the    thumb-nail    or    an   ivory   knife 
handle. 

PROTRACTOR. 

61.  For  laying    out  angles  a  graduated  arc  called  a  "protractor"  is  used.      Various  materials  are 
employed    in    the    manufacture    of  protractors,  Fig-.  31. 

as  metal,  horn,  celluloid,  Bristol  board  and 
tracing  paper.  The  two  last  are  quite  accu- 
rate enough  for  ordinary  purposes,  although 
where  the  utmost  precision  is  required,  one  of 
German  silver  should  be  obtained,  with  a 
moveable  arm  and  vernier  attachment. 

The    graduation,  may    advantageously    be 
to  half  degrees  for  average  work. 

To  lay  out  an  angle  (say  40°)  with  a 
protractor,  the  radius  CH  (Fig.  31)  should 
be  made  to  coincide  with  one  side  of  the  desired  angle;  the  centre,  C,  with  the  desired  vertex; 
and  a  dot  made  with  the  pencil  opposite  division  numbered  40  on  the  graduated  edge.  The  line 
MC,  through  this  point  and  C,  completes  the  construction. 

BRUSHES. 

62.  Sable -hair  brushes  are  the  best  for  laying  flat   or    graduated    tints,  with  ink  or  colors,  upon 
small    surfaces;    while    those    of   camel's    hair,  large,  with    a    brush    at    each    end    of  the    handle,    are 
better   adapted   for   tinting  large   surfaces.      Reject    any    brush    that    does    not    come    to   a  perfect  point 
on   being   moistened.      Five   or  six   brushes   of  different   sizes   are   needed. 

PRELIMINARIES    TO    PRACTICAL    WORK. 

63.  The  first  work  of  a  draughtsman,  like  most  of  his  later  productions,  consists   of  line  as  distin- 
guished from  brush  work,  and  for  it  the  paper  may  be  fastened  upon  the  board  with  thumb-tacks  only. 


UNIVERSITY 
CALIFOP 


20 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


There  is  no  universal  standard  as  to  size  of  sheets  for  drawings.  As  a  rule  each  draughting 
office  has  its  own  set  of  standard  sizes,  and  system  of  preserving  and  indexing.  The  columns  of 
the  various  engineering  papers  present  frequent  notes  on  these  points,  and  the  best  system  of  pre- 
serving and  recording  drawings,  tracings  and  corrections  is  apparently  in  process  of  evolution.  For 
the  student  the  best  plan  is  to  have  all  drawings  of  the  same  size  bound  in  neat  but  permanent 
form  at  the  end  of  the  course.  The  title-pages,  which  presumably  have  also  been  drawn,  will  suf- 
ficiently distinguish  the  different  sets. 

64.  In  his  elementary  work  the  student  may  to  advantage  adopt  two  sizes  of  sheets  which  are 
considerably  employed,  9"  X  13",  and  its  double,  13"  x  18";  sizes  into  which  a  "Super  Royal"  sheet 
naturally  divides,  leaving  ample  margins  for  the  mucilage  in  case  a  "stretch"  is  to  be  made. 

A  "  Double  Elephant "  sheet,  being  twice  the  size  of  a  "  Super  Royal,"  divides  equally  well  into 
plates  of  the  above  size,  but  is  preferable  on  account  of  its  better  quality. 

To  lay  out  four  rectangles  upon  the  paper,  locate  first  the  centre  (see  Fig.  32)  by  intersecting 
diagonals,  as  at  0.  These  should  not  be  drawn  entirely  across  the  sheet,  but  one  of  them  will 
necessarily  pass  a  short  distance  each  side  of  the  point  where  the  centre  lies — judging  by  the  eye 
alone;  the  second  definitely  determines 
the  point.  If  the  T-rule  will  not 
reach  diagonally  from  corner  to  corner 
of  the  paper  (and  it  usually  will  not), 
the  edge  may  be  practically  extended 
by  placing  a  triangle  against  but  pro- 
jecting beyond  it,  as  in  the  upper  left- 
hand  portion  of  the  figure. 

The  T-rule  being  placed  as  shown, 
with  its  head  at  the  left  end  of  the 
board — the  correct  and  usual  position 
—  draw  a  horizontal  line  X  Y,  through 
the  centre  just  located.  The  vertical 
centre  line  is  then  to  be  drawn,  with 
one  of  the  triangles  placed  as  shown 
in  the  figure,  i.  e.,  so  that  a  side,  as  mn  or  tr,  is  perpendicular  to  the  edge. 

It  is  true  that  as  long  as  the  edges  of  the  board  are  exactly  at  right  angles  with  each  other, 
we  might  use  the  T-rule  altogether  for  drawing  mutually  perpendicular  lines.  This  condition  being, 
however,  rarely  realized  for  any  length  of  time,  it  has  become  the  custom  —  a  safe  one,  as  long  as 
rule  and  triangle  remain  "true"  —  to  use  them  as  stated. 

The  outer  rectangles  for  the  drawings  (or  "  plates,"  in  the  language  of  the  technical  school)  are 
completed  by  drawing  parallels,  as  JN  and  Y  N,  to  the  centre  lines,  at  distances  from  them  of  9" 
and  13"  respectively,  laid  off  from  the  centre,  0. 

An  inner  rectangle,  as  abed,  should  be  laid  out  on  each  plate,  with  proper  margins ;  usually 
at  least  an  inch  at  the.  top,  right  and  bottom,  and  an  extra  half  inch  on  the  left  as  an  allowance  for 
binding.  These  margins  are  indicated  by  x  and  z  in  the  figure,  as  variables  to  which  any  con- 
venient values  may  be  assigned.  The  broad  margin  x  in  the  upper  rectangle  will  be  at  the  draughts- 
man's left  hand  if  he  turns  the  board  entirely  around — as  would  be  natural  and  convenient — when 
ready  to  draw  on  the  rectangle  Q  Y. 


EXERCISES    FOR    PEN    AND    COMPASS. 


21 


CHAPTER    IV. 


GRADES    OF    LINES.  — LINE    TINTING.  — LINE    SHADING.  — CONVENTIONAL    SECTION-LINING.— 
FREQUENTLY    RECURRING    PLANE    PROBLEMS.— MISCELLANEOUS    PEN    AND 

COMPASS   EXERCISES. 


65.  Several  kinds  of  lines  employed  in  mechanical  drawing  are  indicated  in  the  figure  below. 
While  getting  his  elementary  practice  with  the  ruling-pen  the  student  may  group  them  as  shown, 
or  in  any  other  symmetrical  arrangement,  either  original  with  himself  or  suggested  by  other  designs. 

Fig.  33. 


FOR  ORDINARY  OUTLINES. 


MEDIUM,     Continuous. 


'DO    T_ED_LI  N  Ejl  u 


"^f1"^" 


SHADE 
LlfME 


/  30^XE 


"DOTTED  LINE'l  line  of  mot 


3n  in  Kinematic  Geometry. 


DIMENSION 


DIMENSION 


.INE,    ifred. 


.INE,    black. 


When  drawing  on  tracing  cloth  or  tracing  paper,  for  the  purpose  of  making  blue -prints,  all  the 
lines  will  preferably  be  black,  and  the  centre  and  dimension  lines  distinguished  from  others  as  indi- 
cated above,  as  also  by  being  somewhat  finer  than  those  employed  for  the  light  outlines  of  the 
object.  Heavy,  opaque,  red  lines  may,  however,  be  used,  as  they  will  blue -print,  though  faintly. 

There  is  at  present  no  universal  agreement  among  the  members  of  the  engineering  profession  as 
to  standard  dimension  and  centre  lines.  Not  wishing  to  add  another  to  the  systems  already  at 
variance,  but  preferring  to  facilitate  the  securing  of  the  uniformity  so  desirable,  I  have  presented 
those  for  some  time  employed  by  the  Pennsylvania  Railroad  and  now  taught  at  Cornell  University. 

The  lines  of  Fig.  33,  as  also  of  nearly  all  the  other  figures  of  this  work,  having  been  printed  from  blocks  made  by  the 
cerographie  process  (Art.  277),  are  for  the  most  part  too  light  to  serve  as  examples  for  machine-shop  work.  Fig.  80  is  a  sample. 
of  P.  R.  R.  drawing,  and  is  a  fair  model  as  to  weight  of  line  for  working  drawings. 


22 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


A  dash  -and  -three  -dot  line  (not  shown  in  the  figure)  is  considerably  used  in  Descriptive  Geometry, 
either  to  represent  an  auxiliary  plane  or  an  invisible  trace  of  any  plane.  (See  Fig.  238). 

The  so-called  "dotted"  line  is  actually  composed  of  short  dashes.  Its  use  as  a  "line  of 
motion  "  was  suggested  at  Cornell. 

When  colors  are  used  without  intent  to  blue  -print  they  may  be  drawn  as  light,  continuous  lines. 
Colors  will  further  add  to  the  intelligibility  of  a  drawing  if  employed  for  construction  lines.  Even 
if  red  dimension  lines  are  used,  the  arrow  heads  should  invariably  be  black.  They  should  be  drawn 
free-hand,  with  a  writing  pen,  and  their  points  touch  the  lines  between  which  they  give  the  distance. 

66.  The  utmost  accuracy  is  requisite  in  pencilling,  as  the  draughtsman  should  be  merely  a  copyist 
when  using  the  pen.  On  a  complicated  drawing  even  the  kind  of  line  should  be  indicated  at  the 
outset,  so  that  no  time  will  be  wasted,  when  inking,  in  the  making  of  distinctions  to  which  thought 
has  already  been  given  during  the  process  of  construction.  No  unnecessary  lines  should  be  drawn, 
or  any  exceeding  of  the  intended  limit  of  a  line  if  it  can  possibly  be  avoided. 

If  the  work  is  symmetrical,  in  whole  or  in  part,  draw  centre  lines  first,  then  main  outlines  ;  and 
continue  the  work  from  large  parts  to  small. 

The   visible   lines   of  an   object  are  to  be  drawn  first;    afterward  those  to  be  indicated  as  concealed. 

All  lines  of  the  same  quality  may  to  advantage  be  drawn  with  one  setting  of  the  pen,  to  ensure 
uniformity;  and  the  light  outlines  before  the  shade  lines. 

In   drawing  arcs   and   their  tangents,  ink  the   former  first,   invariably. 

All  the  inking  may  best  be  done  at  once,  although  for  the  sake  of  clearness,  in  making  a  large 
and  complicated  drawing,  a  portion  —  usually  the  nearest  and  visible  parts  —  may  be  inked,  the  draw- 
ing cleaned,  and  the  pencilling  of  the  construction  lines  of  the  remainder  continued  from  that  point, 

The    inking    of   the    centre,   dimension    and    construction  lines   naturally   follows    the    completion   of 


the   main   design. 


2     |    3 


|  a  | 


67.  In  Fig.  34  we  have  a  straight -line  design 
usually  called  the  "  Greek  Fret,"  and  giving  the 
student  his  first  illustration  of  the  use  of  the 
"shade  line"  to  bring  a  drawing  out  "in  relief." 
The  law  of  the  construction  will  be  evident  on 
examination  of  the  numbered  squares. 

Without  entering  into  the  theory  of  shadows  at 
this  point,  we  may  state  briefly  the  "  shop  rule " 
for  drawing  shade  lines,  viz.,  right-hand  and  lower. 
That  is,  of  any  pair  of  lines  making  the  same 
turns  together  or  representing  the  limit  of  the  same 


as. 


flat  surface,  the  right  -hand  line  is  the   heavier  if  the  pair 
is  vertical,  but  the  lower  if  they  run  horizontally;   always 
subject,    however,    to  the    proviso  that  the  line  of   inter- 
section  of   two    illumin- 
ated   planes    is   never    a 
shade  line. 

68.  The  conic  section 
called  the    parabola  fur- 

nishes another    interesting    exercise    in    ruled  lines,  when 
it    is   represented    by    its    tangents    as    in  Fig.  35.      The    angle   CAE 
may    be    assumed  at   pleasure,  and    on   the  finished  drawing  the  numbers  may 


THE  PARABOLA 

BY 
ENVELOPING  TANGENTS. 


SECTION- LINING. —  LINE-SHADING. 


23 


-.  OS. 


be   omitted,  being    given  here  merely    to    show    the    law    of   construction.     All  the  divisions  are  equal, 
and  like   numbers   are  joined. 

Some   interesting  mathematical   properties   of  the   curve   will   be   found   in   Chapter  V. 

69.  A  pleasing  design  that  will  test   the  beginner's  skill  is  that    of   Fig.  36.      It  is  suggestive  of 
a  cobweb,  and  a  skillful  free-hand    draughtsman  could  make  it  more  realistic   by  adding   the    spider. 
Use    the    60°   triangle    for    the    heavy    diagonals 

and  parallels  to  them;  the  T-rule  for  the  hor- 
izontals. Pencil  the  diagonals  first  but  ink  them 
last. 

70.  The    even  or  flat    effect    of   equidistant 
parallel  lines  is  called  line  -  tinting  ;    or,  if  repre- 
senting an  object  that  has  been  cut  by  a  plane, 
as   in   Fig.  37,   it  is   called  section  -lining. 

The  section,  strictly  speaking,  is  the  part 
actually  in  contact  with  the  cutting  plane; 
while  the  drawing  as  a  whole  is  a  sectional 
view,  as  it  also  shows  what  is  back  of  the 
plane  of  section,  the  latter  being  always  as- 
sumed to  be  transparent. 


Adjacent    pieces 
have  the  lines  drawn 
J3    in  different  directions 

in  order  to  distinguish  sufficiently  between  them. 

The  curved  effect  on  the  semi -cylinder  is  evidently  obtained  by  prop- 
erly  varying   both   the   strength   of  the  line  and  the  spacing. 

71.  The  difference  between  the  shading  on  the  exterior  and  interior 
H  of  a  cylinder  is  sharply  contrasted  in  Fig.  38.  On  the  concavity  the 
darkest  line  is  at  the  top,  while  on  the  convex  surface  it  is  near  the  bot- 
tom, and  below  it  the  spaces  remain  unchanged  while  the  lines  diminish. 
A  better  effect  would  have  been  obtained  in  the  figure  had  the  engraver  begun  to  increase  the  lines 
with  the  first  decrease  in  the  space  between  them. 

•-  33. 


37. 


w 


The  spacing  of  the  lines,  in  section -lining,  depends  upon  the  scale  of  the  drawing.  It  may  run 
down  to  a  thirtieth  of  an  inch  or  as  high  as  one -eighth;  but  from  a  twentieth  to  a  twelfth  of  an 
inch  would  be  best  adapted  to  the  ordinary  range  of  work.  Equal  spacing  and  not  fine  spacing 


24 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


should  be  the  object,  and    neither    scale    nor    patent    section-liner    should    be   employed,  but  distances 
gauged  by  the  eye  alone. 

72.  A    refinement    in    execution    which    adds    considerably    to    the    effect  is  to  leave  a  white  line 
between    the    top    and    left-hand  outlines    of   each    piece    and    the    section  lines.      When  purposing    to 
produce   this   effect,  rule  light   pencil  lines   as  limits   for   the   line -tints. 

73.  If  the    various    pieces    shown    in    a    section  are  of  different  materials,  there  are  four  ways   of 
denoting  the   difference  between   them : 

(a)  By  the  use  of  the  brush  and  certain  water -colors,  a  method  considerably  employed  in  Europe, 
but  not  used  to  any  great  extent  in  this  country,  probably  owing  to  the  fact  that  it  is  not  applic- 
able where  blue -prints  of  the  original  are  desired. 

The  use  of  colors  may,  however,  be  advantageously  adopted  when  making  a  highly  finished, 
shaded  drawing;  the  shading  being  done  first,  in  India  ink  or  sepia,  and  then  overlaid  with  a  flat 
tint  of  the  conventional  color.  The  colors  ordinarily  used  for  the  metals  are 

Payne's  gray  or  India  ink  for  Cast  Iron. 

Gamboge  "     Brass   (outside  view). 

Carmine  "     Brass  (in  section). 

Prussian  Blue  "     Wrought  Iron. 

Prussian  Blue  with  a  tinge  of  Carmine    "     Steel. 


Cast   Iran. 


StEEl. 


Wr't.    Iran. 


Brass. 


Sectiens, 


StnriE. 


Wood. 


Cap  PET. 


Brick. 


CONVENTIONAL    SECTION-LINING. 


25 


More  natural  effects  can  also  be  given  by  the  use  of  colors,  in  representing  the  other  materials 
of  construction;  and  the  more  of  an  artist  the  draughtsman  proves  to  be,  the  closer  can  he  approx- 
imate to  nature. 

Pale  blue  may  be  used  for  water  lines;  Burnt  Sienna,  whether  grained  or  not,  suggests  wood; 
Burnt  Umber  is  ordinarily  employed  for  earth;  either  Light  Red  or  Venetian  Red  are  well  adapted 
for  brick,  and  a  wash  of  India  ink  having  a  tinge  of  blue  gives  a  fair  suggestion  of  masonry; 
although  the  actual  tint  and  surface  of  any  rock  can  be  exactly  represented  after  a  little  practice 
with  the  brush  and  colors.  These  points  will  be  enlarged  upon  later. 

(b)  By  section  -  lining   with   the   drawing   pen   in   the   conventional   colors   just   mentioned,   a   process 
giving    very    handsome    and    thoroughly    intelligible    results    on    the    original    drawing,   but,    as    before, 
unadapted  to   blue -printing   and   therefore   not   as   often   used   as   either   of  the   following   methods. 

(c)  By   section -lining   uniformly   in  ink  throughout,  and   printing  the    name    of   the   material   upon 
each   piece. 

(d)  By  alternating  light  and  heavy,   continuous   and  broken 
lines,  according  to   some  law.      Said  "  law "   is,  unfortunately,  by 
no   means   universal,  despite  the   attempt  made  at  a  recent  con- 
vention   of   the    American    Society    of    Mechanical    Engineers   to 
secure   uniformity.      Each   draughting   office   seems   at  present  to 
be  a  law   unto   itself  in  this   matter. 

74.  As  affording  valuable  examples  for  further  exercise 
with  the  ruling  pen,  the  system  of  section -lines  adopted  by 
the  Pennsylvania  Railroad  is  presented  on  the  opposite  page. 
The  wood  section  is  an  exception  to  the  rule,  being  drawn 
free-hand,  with  a  Falcon  pen. 

By  way  of  contrasting  free-hand  with  me- 
chanical work  Fig.  40  is  introduced,  in  which 
the  rings  showing  annual  growth  are  drawn 
as  concentric  circles  with  the  compass. 

In  Fig.  41  a  few  other  sections  appear, 
selected  from  the  designs  of  M.  N.  Forney 


and   F.   Van   Vleck,   and   which   are   fortunate  arrangements. 

75.  Figs.  42  and  43  are  profiles  or  outlines  of  mouldings, 
such  as  are  of  frequent  occurrence  in  architectural  work.  It  is 
good  practice  to  convert  such  views  into  oblique  projections,  giving  the  effect  of  solidity;  and  to 
further  bring  out  their  form  by  line  shading.  Figs.  44-46  are  such  representations,  the  front  of  each 
being  of  the  same  form  as  Fig.  42.  The  oblique  lines  are  all  parallel  to  each  other,  and — where 


visible  throughout — of  the  same  length.  Their  direction  should  be  chosen  with  reference  to  best  ex- 
hibiting the  peculiar  features  of  the  object.  Obviously  the  view  in  Fig.  44  is  the  least  adapted  to 
the  conveying  of  a  clear  idea  of  the  moulding,  while  that  of  Fig.  46  is  evidently  the  best. 


26 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


76.     The    student    may,  to    advantage,   design    profiles    for    mouldings    and    line  -shade    them,    after 
converting   them   into   oblique   views.      As    hints    for    such    work    two    figures    are  given   (47-48),   taken 


-.  -US- 


from    actual    construction    in   wood.      By    setting    a    moulding    vertically,  as  in  Fig.  49,  and  projecting 
horizontally  from  its  points,  a  front  view  is  obtained,  as  in   Fig.  50. 


Flgf. 


--  -ie. 


.  SO. 


77.    The    reverse    curves    on    the    mouldings    may  be  drawn  with  the  irregular  curve,   (see  Art  58); 


r-  si. 


M 


or,  if  composed  of  circular  arcs  to  be  tangent  to  vertical    lines,  by  the  follow- 
ing  construction :  - — 

Let  M  and  N  be  the  points  of  tangency  on  the  verticals  Mm  and  Nn, 
and  let  the  arcs  be  tangent  to  each  other  at  the  middle  point  of  the  line 
MN.  Draw  Mn  and  Nm  perpendicular  to  the  vertical  lines.  The  centres,  c 
and  cn  of  the  desired  arcs,  are  at  the  intersection  of  Mn  and  Nm  by  per- 
pendiculars to  M  N  from  x  and  y,  the  middle  points  of  the  segments  of  MN. SL 

78.     The    light    is    to    be    assumed    as    coming    in  the   usual   direction,   i.  e., 
descending   from   left  to   right   at  such   an  angle   that  any   ray   would   be   projected   on   the  paper  at  an 
angle  of  45°  to  the  horizontal. 

In  Fig.  43  several  rays  are  shown.  At  z,  where  the  light  strikes  the  cylindrical  portion  most 
directly — technically  is  normal  to  the  surface— is  actually  the  brightest  part.  A  tangent  ray  st  gives 
t,  the  darkest  part  of  the  cylinder.  The  concave  portion  beginning  at  o  would  be  darkest  at  o  and 
get  lighter  as  it  approaches  y. 

Flat  parts  are  either  to  be  left  white,  if  in  the  light,  or  have  equidistant  lines  if  in  the  shade, 
unless  the  most  elegant  finish  is  desired,  in  which  case  both  change  of  space  and  gradation  of  line 

must  be  resorted  to  as  in  Fig.  52,  which  represents  a  front  view  of  a 
hexagonal  nut.  The  front  face,  being  parallel  to  the  paper,  receives  an 
even  tint.  An  inclined  face  in  the  light,  as  abhf,  is  lightest  toward 
the  observer,  while  an  unillumined  face  tkdg  is  exactly  the  reverse. 

Notice  that  to  give  a  flat  effect  on  the  inclined  faces  the  spacing - 
out   as   also   the  change   in  the  size  of  lines  must  be  more  gradual  than 
a     when   indicating   curvature.      (Compare   with   Figs.  46  and  50.) 


-.  S2. 


rr 

REMARKS    ON    SHADING  .  —  PL  ANE    PROBLEMS.  27 

If  two  or  more  illuminated  flat  surfaces  are  parallel  to  the  paper  (as  t  g  b  h,  Fig.  52)  but  at 
different  distances  from  the  eye,  the  nearest  is  to  be  the  lightest;  if  unilluminated,  the  reverse 
would  be  the  case. 

79.  In  treating  of  the  theory  of  shadows,  distinctions  have  to  be  made,  not  necessary  here,  between 
reed  and    apparent  .  brilliant   points   and   lines.      We   may   also   remark   at   this   point    that    to    an    experi- 
enced   draughtsman    some    license    is    always    accorded,   and    that    he    can    not    be    expected    to    adhere 
rigidly    to    theory    when    it    involves   a   sacrifice   of  effect.      For   example,   in   Fig.  46   we   are   unable   to 
see  to  the  left  of  the  (theoretically)   lightest   part    of   the  cylinder,   and  find  it,  therefore,  advisable   to 
move    the    darkest    part    past    the    point    where,   according    to    Fig.  43,   we    know    it    in    reality   to    be. 
The  professional  draughtsmen   who  draw  for  the  best  scientific  papers,  and    to    illustrate    the    circulars 
of  the   leading   machine   designers,  allow   themselves   the   latitude   mentioned,   with   most   pleasing   results. 
Yet   until   one  may   be    justly    called   an   expert   he   can   depart    but   little  from   the   narrow   confines    of 
theory    without   being  in   danger   of  producing   decidedly   peculiar  effects. 

80.  As   from   this   point  the   student   will  make   considerable   use    of   the   compasses,   a    few    of   the 
more  important  and  frequently  recurring  plane  problems,  nearly  all  of  which  involve  their  use,  may  well 
be  introduced.     The  proofs  of  the  geometrical  constructions  are  in  several  cases  omitted,  but  if  desired 
the  student  can  readily  obtain  them  by  reference  to  any  synthetic  geometry  or  work  on  plane  problems. 

All  the  problems  given  (except  No.  20)  have  proved  of  value  in  shop  practice  and  architectural  work. 

The  student  should  again  read  Arts.  48-51  regarding  special  uses  of  the  30°  and  45°  triangles, 
which,  with  the  T-  rule,  enable  him  to  employ  so  many  "  draughtsman's  "  as  distinguished  from 
"geometrician's"  methods;  also  Arts.  36  and  37. 

81.  Prob.  1.     To   draw  a  perpendicular    to  a  given  line  at  a  given  point,  as  A  (Fig.   53),   use  the  tri- 
angles, or  triangle  and  rule  as  previously  described  ;   or  lay  off  equal  distances  A  a,  Ab,  and  with  a  and 
6  as  centres  draw  arcs  ost,  msn,  with  common  radius  greater  than  one  -half  a  b.     The  required  perpen- 


dicular is  the  line  joining  A  with  the  intersection  of  these  arcs.  ^igr-  53. 


82.  Prob.  2.     To    bisect    a    line,   as   MN,    use    its    extremities 
exactly  as  a  and  6   were  employed  in  the  preceding  construc- 
tion,  getting    also    a    second   pair  of  arcs    (same  radius  for  all 
the  arcs)  intersecting  above  the  line  at  a  point  we  may  call  x. 

The   line   from   s  to   x  will   be   a  bisecting   perpendicular.  m. — -^^       ~~~~ 1 

83.  Prob.  S.     To  bisect  an  angle,  as  A  VB,  (Fig.  54),  lay   off  on  its  sides  any  equal  distances   V  a, 

&&-  Vb.      Use    a    and   b    as    centres    for    intersecting    arcs    having    a 

common  radius.      Join    V  with  x,  the  intersection  of  these   arcs, 
for  the   bisector   required. 

84.     Prob.  4-     To   bisect  an  arc  of  a  circle,  as   amb  (Fig.  54), 
bisect    the    chord    a  c  b    by    Prob.  2 ;     or,   by    Prob.  3,   bisect    the 
angle  a  V  b  which  subtends  the  arc. 
85.     Prob.  5.     To    construct    an    angle  equal  to   a  given   angle,  as   6    (Fig.  55),  draw   any  arc   a  b   with 
centre   0,  then,    with    same    radius,    an    indefinite    arc    m  B,  E-IS--  SB. 

centre  V ;  use  the  chord  of  a  b  as  a  radius,  and  from 
centre  B  cut  the  arc  m  B  at  x.  Join  V  and  x.  Then 
angle  AVB  equals  6. 

86.  Prob.  6.  To  pass  a  circle  through  three  points,  a,  b 
and  c,  join  them  by  lines  a  b,  be,  bisect  these  lines  by 
perpendiculars,  and  the  intersection  of  the  latter  will  be  the  centre  of  the  desired  circle. 


28 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


Fig.  SS. 


B  \ 


87.     Prob.  7.      To    divide    a    line    into    any    number    of   equal   parts,  draw    from    one    extremity,  as  A, 

(Fig.  56),  a  line  A  C  making  any  random  angle  with  the  given  line 
A  B.  With  a  scale  point  off  on  A  C,  as  many  equal  parts  (size 
immaterial)  as  are  required  on  A  B ;  four,  for  example.  Join  the 
last  point  of  division  (4)  with  B;  then  parallels  to  such  line  from 
the  other  points  will  divide  A  B  similarly. 

88.  A  secant  to  a  curve  is  a  line  cutting  it  in  two  points.  If 
the  secant  A  B  be  turned  to  the  left  about  A,  the  point  B  will  approach  A,  and  the  line  will  pass 
through  A  C  and  other  secant  positions.  When  B  reaches  and  coincides  E'lg-.  S7. 

with  A   the  line  is  said  to  be  tangent  to  the  curve.     (See  also  Art.  368.) 

A    tangent    to    a    mathematical    curve    is    determined    by    means    of    known 
properties  of  the   curve.      For  a  random  or  graphical  curve   the  method   illus- 
trated by  Fig.  57  (a)    is  the  most  accurate  and  is  as  follows:     Through    T,  the  point   of   desired  tan- 
.  &?  (a)  gency,  draw    random    secants    to    points    on  either  side    of   it,  as  A,  B,  D,   etc., 

and  prolong  them  to  meet  a  circle  having  centre  T  and  any  radius.  On  each 
secant  lay  off — from  its  intersection  with  the  circle — the  chord  of  that  secant 
in  the  random  curve.  Thus,  am=TA;  bn=TB;  pd=TD.  From  s 
where  the  curve  m  n  o  p  q  cuts  the  circle,  draw  s  T,  which  will  be  a  tangent, 
since  for  it  the  chord  has  its  minimum  value. 

A    normal    to    a    curve    is  a   line   perpendicular  to   the   tangent,   at   the   point 
of  tangency.      In   a  circle   it  coincides   in   direction   with   the  radius   to  the  point   of  tangency. 

89.  Prob.  8.      To    draw    a    tangent    to    a    circle  at  a  given  point  draw   a   radius   to   the   point.      The 
perpendicular  to  this   radius   at  its   extremity   will   be   the   required  tangent.      Solve   with   triangles. 

90.  Prob.  9.      To    draw  a  tangent    to   a   circle  from  a   point  without, 
join   the  centre   C  (Fig.  58)   with   the  given   point  A ;    describe  a  semi- 
circle on  A  C  as  a  diameter  and   join  A   with  D,  the  intersection  of 
the  arcs.      ADC   equals   90°,   being    inscribed    in   a   semi -circle;     AD 
is    then    the    required    tangent,   being    perpendicular    to    CD    at    its 
extremity. 

91.  Prob.  10.      To    draw  a  tangent  at    a    given   point    of  a    circular  ^ '      ^igr-  se. 

arc    whose    centre    is    unknown    or    inaccessible,    locate    on   the  arc    two   points    equidistant    from    the    given 
point  and  on  opposite  sides  of  it;    the  chord  of  these  points  will  be    parallel  to  the  tangent  sought 

92.  A  regular  polygon   has   all  its   sides   equal,   as   also  its   angles.     If    of   three    sides    it    is   called 
the  equilateral    triangle;    four    sides,   the    square;     five,    pentagon;     six,   hexagon;     seven,   heptagon;     eight, 
octagon;     nine,   nonagon  or  enneagon;     ten,   decagon;     eleven,   undecagon ;     twelve,   dodecagon. 

The  angles  of  the  more  important  regular  polygons  are  as  follows :  triangle, 
120°;  square,  90°;  pentagon,  72°;  hexagon,  60°;  octagon,  45°;  decagon,  36°; 
dodecagon,  30°.  The  angle  at  the  vertex  of  a  regular  polygon  is  the  supplement 
of  its  central  angle. 

93.  For  the  polygons  most  frequently  occurring  there  are  many  special 
methods  of  construction.  All  but  the  pentagon  and  decagon  can  be  readily 
inscribed  or  circumscribed  about  a  circle  by  the  use  of  the  T-rule  and  triangle. 
For  example,  draw  a  b  (Fig.  59)  with  the  T-  rule,  and  c  d  perpendicular  to  it 
with  a  triangle.  The  45  °  triangle  will  then  give  a  square,  a  c  b  d.  The  same  triangle  in  two  positions 
would  give  ef  and  g  h,  whence  ag,  g  c,  etc.,  would  be  sides  of  a  regular  octagon. 


f 


PLANE    PROBLEMS. 


29 


94.  The  60°  triangle  used    as    in  Art.  51   would  give  the  hexagon;    and    alternate  vertices  of  the 
latter,  joined,  would  give  an  equilateral  triangle.    Or  the  radius  of   the  circle  stepped  off  six  times  on 
the   circumference,   and   alternate   points   connected,   would   result  similarly. 

95.  Prob.  11.     An    additional    method    for    inscribing    an    equilateral    triangle    in    a 
circle,  when  one  vertex  of  the  triangle  is  given,  as  A,  Fig.  60,   is  to   draw   the  diameter, 
A  B,  through  A,  and    use    the    triangle    to    obtain    the    sides  A  0   and  A  D,   making 
angles  of   30  °   with  A  B.      D  and   C  will  then  be  the  extremities   of   the  third  side 
of  the  triangle  sought. 

96.      Prob.    12.      To    inscribe    a    circle    in    an    equilateral    triangle, 

draw  a  perpendicular  from  any  vertex  to  the  opposite  side.  The  centre  of  the 
circle  will  be  on  such  line,  two -thirds  of  the  distance  from  vertex  to  base,  while 
the  radius  desired  will  be  the  remaining  third.  (Fig.  61). 

97.  Prob.  IS.  To  inscribe  a  circle  in  any  triangle,  bisect  any  two  of  the  interior 
angles.  The  intersection  of  these  bisectors  will  be  the  centre,  and  its  perpen- 
dicular distance  from  any  side  will  be  the  radius  of  the  circle  sought. 

98.  Prob.  14.      To  inscribe  a  pentagon  in  a  circle,  draw   mutually   perpendicular 
diameters    (Fig.  62);     bisect    a    radius    as    at    s;     draw    arc   a  a;    of    radius   sa   and 
centre   s;    then   chord   ax=af,   the   side   of  the   pentagon   to   be   constructed. 

99.  Prob.  15.      To  construct  a  regular  polygon  of  any  number  of  sides,   the  length 
of  the  side  being  given. 

Let  A  B  (Fig.  63)  be  the  length  assigned  to  a  side,  and  a  regular  polygon 
of  x  sides  desired.  Take  x  equal  to  nine  for  illustration,  draw  a  semi -circle  with 
A  B  as  radius,  and  divide  by  trial  into  x  (or  9)  equal  parts.  Join  B  with  x —2 

points  of  division,  or  seven,  beginning  at  A,  and  prolong  all  but  the 
last.  With  7  as  a  centre,  radius  A  B,  cut  line  B-6  at  m  by  an 
arc,  and  join  m  with  7,  giving  another  side  of  the  required  polygon. 
Using  m  in  turn  as  a  centre,  same  radius  as  before,  cut  B-5  (pro- 
duced) and  so  obtain  a  third  vertex. 

This    solution    is    based   on   the   familiar   principles    (a)    that  if   a 
regular    polygon    has   x 
sides,  each  interior  angle 

180°  (x 2^ 

equals    -  — '-,    and    (b)    that    the    diagonals    drawn 

cc 

from   any   vertex    of   the    polygon    make  the   same   angles 
with  each  other  as  with  the  sides  meeting  at  that  vertex. 

100.  Prob.  16.     Another  solution  of  Prob.  15.     Erect  a 
perpendicular  HR  (Fig.  64)    at  the  middle  point  of  the 
given    side.      With  M  as    a    centre,    radius  MS,    describe 
arc  SA  and  divide  it  by  trial  into  six  equal  parts.     Arcs 
through    these    points    of   division,   using   A    as    a    centre, 
and  numbered  up  from  six,  give  the  centres  on  the  ver- 
tical line  for   circles    passing    through   M  and    S,  and    in 
which    MS   would    be    a    chord    as    many    times    as    the 
number  of  the   centre. 

101.  For  any   unusual   number    of   sides   the   method  ^B!^=SSS^  FIS.  s-i. 


, 
\ 


30 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


PLANE    PROBLEMS. 


81 


by  "trial  and  error"  is  often  resorted  to,  and  even  for  ordinary  cases  it  is  by  no  means  to  be 
despised.  By  it  the  dividers  are  set  "  by  guess "  to  the  probable  chord  of  the  desired  arc,  and,  sup- 
posing a  heptagon  wanted,  the  chord  is  stepped  off  seven  times  around  the  circumference;  care  being 
taken  to  have  the  points  of  the  dividers  come  exactly  on  the  arc,  and  also  to  avoid  damaging  the 
paper.  If  the  seventh  step  goes  past  the  starting  point,  the  dividers  require  closing;  if  it  falls  short, 
the  original  estimate  was  evidently  too  small.  Obviously,  the  change  in  setting  the  dividers  ought 
in  this  case  to  be,  as  nearly  as  possible,  one- seventh  of  the  error;  and  after  a  few  trials  one  should 
"  come  out  even "  on  the  last  step. 

102.  Prob.  17.     To  lay  off  on  a  given  circle  an  arc  of  the  same  length  as  a  given  straight  line.1    Let 
t  (Plate   I,  Fig.  1)  be  one  extremity  of   the  desired  arc;    ts   the    given    straight   line    and    tangent  to 
the   circle;     tm    equal    one -fourth   of   ts,  and    sx  drawn   with   centre  m,   radius    ms.      Then  the  length 
of  the  arc  tx  is  a  close  approximation  to  that  of  the  line  ts. 

103.  Prob.  18.     To  lay  off   on    a    straight    line    the  length  of   a  given  circular  arc,1   or,   technically,  to 
rectify  the  arc,  let  af  (Plate  I,  Fig.  3)  be  the  given  arc;    ai  the  chord  prolonged  till  fi  equals  one- 
half  the  chord  af;    and  ae  an  arc  drawn  with  radius  ai,  centre  i.      Then  fe   approximates    closely 
to  the  length  of  the  arc  af. 

104.  Prob.  19.     To  obtain  a  straight  line  equal  in  length  to  any  given  semi- circle?  draw  a  diameter  oh 
of  the  given  semi -circle  (Plate  I,  Fig.  2)  and  a  radius  inclined  at  an  angle  of  30°  to  the  radius  ch. 
Prolong  the  radius  to  meet  the   line  b  h  k,  drawn  tangent    to    the    circle    at    h.      From    k    lay    off  the 
radius  three  times,  reaching  n.     The  line  no  equals  the  semi -circumference  to  four  places  of  decimals. 

105.  Prob.  20.      To  draw  a  circle  tangent  to   two  straight  lines  and    a  given  circle.      (Four    solutions.) 
This  problem  is  given  more  on  account  of   the  valuable  exercise  it  will  prove  to  the  student  in  ab- 
solute precision  of   construction    than    for   its    probable    practical    applications.      Fig.  4   (Plate  I)   illus- 
trates the  geometrical  principles  involved,  and  in  it  a  circle  is  required  to  contain  the  points    s    and 


i  These  methods  of  approximation  were  devised  by  Prof.  Kankine.  They  are  sufficiently  accurate  for  arcs  not  exceeding 
60°.  The  error  varies  as  the  fourth  power  of  the  angle.  The  complete  demonstration  of  Prob.  17  can  be  found  in  the  Philo- 
sophical Magazine  for  October,  1867,  and  of  Prob.  18  in  the  November  issue  of  the  same  year. 

*In  his  Graphical  Statics  Cremona  states  this  to  be  the  simplest  method  known  for  rectifying  a  semi-circumference.  Accord- 
ing to  Bottcher  it  is  due  to  a  Polish  Jesuit,  Kochansky,  and  was  published  in  the  Acta  Eruditorum  Lipsiae,  1685.  The  demon- 
stration is  as  follows :  Calling  the  radius  unity,  the  diameter  would  have  the  numerical  value  2. 

Then  In  Fig.  2,   Plate   I,  we  have  m  =  •v/oA"  +  An«  =  \/oW  +  (*»  —  khf  =  x/4  +  (3  —  tan  30°)2=  3.14159  + 

The  tangent  of  an  angle  (abbreviated  to  "tan.")  is  a  trigonometric  function  whose  numerical  value  can  be  obtained  from 
a  table.  A  draughtsman  has  such  frequent  occasion  to  use  these  functions  that  they  are  given  here  for  reference,  both  as  lines 
and  as  ratios. 

Trigonometric  Functions  as  Ratios.  Trigonometric   Functions  as  Lines. 


e  =  the  given  angle  —  CAB 

h  =  hypothenuse  of  triangle  CAB 

a  =  A  B  =  side  of  triangle  adjacent  to  vertex  Of  0 

o  =  B  C  =  side  of  triangle  opposite  to  fl 


Cu-tangmt  af  Q 


Then   sin  9  =  j ;  cos 
sin  0 


ton  9  =  5  = 

h 


'  cos  6  ' 

sec  «  =  -J  =  reciprocal  of  cosine. 
A 


cosec  8  = 


sine 


cotan  6 


a      cos  0 

-  =  ggj-g  =  reciprocal  of  tan  0. 


B 


The  prefix  "co"  suggests  "complement;"  the  co-sine  of  0  Is  the  sine  of  the  complement  of  6,  Ac.  As  lines  the  functions 
may  be  defined  as  follows  : 

The  sine  of  an  arc  (e.  g.,  that  subtended  by  angle  6  in  the  figure)  is  the  perpendicular  (C  B)  let  lall  from  one  extremity  of 
the  arc  upon  the  diameter  passing  through  the  other  extremity.  If  the  radius  A  C,  through  one  extremity  of  the  arc,  be 
prolonged  to  cut  a  line  tangent  at  the  other  extremity,  the  intercepted  portion  of  the  tangent  is  called  the  tangent  of  the  arc, 
and  the  distance,  on  such  extended  radius,  from  the  centre  of  the  circle  to  the  tangent,  Is  called  the  secant  of  the  arc. 

The  co-sine,  co-secant  and  co-tangent  of  the  arc  are  respectively  the  sine,  secant  and  tangent  of  the  complement  of  the 
given  arc. 


32 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


a  and  be  tangent  to  the  line  m  vr  Draw  first  any  circle  containing  s  and  a,  as  the  one  called  "  aux. 
circle."  Join  s  to  a  and  prolong  to  meet  mvl  at  k.  From  k  draw  a  tangent,  kg,  to  the  auxiliary 
circle.  With  radius  k  g  obtain  m  and  i  on  the  line  m  v.  A  circle  through  s,  a  and  m,  or  through 
8,  a  and  i  will  fulfill  the  conditions.  For  k  g 2  =  k  s  x  k  a,  as  i  g  is  a  tangent  and  k  s  a  secant. 
But  k  i  =  k  g,  therefore  k  i 2  =  k  s  X  k  a,  which  makes  k  i  a  tangent  to  a  circle  through  s,  a  and  i. 

In  Fig.  5  (Plate  I)  the  construction  is  closely  analogous  to  the  above,  and  the  lettering  identi- 
cal for  the  first  half  of  the  work.  The  "given  circle"  is  so  called  in  the  figure;  the  given  lines 
are  P  v  and  R  v.  Having  drawn  the  bisector,  v  e,  of  the  angle  P  v  R,  locate  s  as  much  below  v  e  as 
a  (the  centre  of  the  given  circle)  is  above  it,  the  line  as  being  perpendicular  to  v  e.  Draw  vlmki 
parallel  to  v  p  and  at  a  distance  from  it  equal  to  the  radius  of  the  given  circle.  Then  s,  a,  k  and 
mvl  of  Fig.  5  are  treated  exactly  as  the  analogous  points  of  Fig.  4,  and  a  circle  obtained  (centre  eZ) 
containing  a,  s  and  i.  The  required  circle  will  have  the  same  centre  d,  but  radius  d  w,  shorter  than 
the  first  by  the  distance  iv  i.  Treat  s,  a,  and  m,  (Fig.  5),  similarly,  getting  the  smallest  of  the  four 
possible  circles. 

The  remaining  solutions  are  obtained  by  using  the  points  a  and  s  again,  but  in  connection  with 
a  line  y  z  parallel  to  v  R  and  inside  the  angle,  again  at  a  perpendicular  distance  from  one  of  the 
given  lines  equal  to  the  radius  of  the  given  circle.* 

This  problem  makes  a  handsome  plate  if  the  given  and  required  lines  are  drawn  in  black;  the 
lines  giving  the  first  two  solutions  in  red;  the  remaining  construction  lines  in  blue.  • 

106.  Prob.  21.     To   draw   a  tangent  to   two  given  circles  (a    problem    that    may    occur    in    connecting 

-.  &7.  band-wheels   by   belts)    join    their    centres,   c    and    o,    (Fig.   67) 

and  at  s  lay  off  s  m  and  s  n  each  equal  to  the  radius  of  the 
smaller  circle.  Describe  a  semi-circle  o  h  k  c  on  o  c  as  a 
diameter.  Carry  m  and  n  to  k  and  h,  about  o  as  a  centre. 
Angles  c  k  o  and  c  h  o  are  each  90  °,  being  inscribed  in  a 
semi-circle ;  and  c  k  is  parallel  to  a  b,  which  last  is  one  of 
the  required  tangents ;  while  c  h  is  parallel  to  t  x,  a  second 
tangent.  Two  more  can  be  similarly  found. 

107.  Prob.  22.     To  unite  two  inclined  straight  lines  by   an  arc  tangent  to   both,  radius  given.      Prolong 

.  ©s-  the    given    lines    to    meet    at    a    (Fig.    68).      With    a  as    a    centre,  and 

the  given  radius,  describe  the  arc  m  n.  Parallels  to  the  given  lines  and 
tangent  to  arc  m  n  meet  at  d,  from  which  perpendiculars  to  the  given 
lines  give  the  points  of  tangency  of  the 
required  arc,  which  is  now  drawn  with 
the  given  radius. 

108.     Prob.   28.      To     draw     through    a    C 
given  point   a   line    which    will — if  produced — pass    through    the     inaccessible 


/ 

Fig-.   ©S. 

h 

*  N 

\\  fr 

B 

3        >/ 

i       / 

*This  solution  is  taken  from  Benjamin  Alvord's  Tangerines  of  Circles  and  of  Spheres,  published  by  the  Smithsonian  Institute. 
That  valuable  pamphlet  presents  geometrical  solutions  of  the  ten  problems  of  Apollonius  on  the  tangencies  of  circles,  and  also 
of  the  fifteen  problems  on  the  tangencies  of  spheres,  all  of  which  are  valuable  to  the  draughtsman,  both  geometrically  and  as 
exercises  in  precision.  The  solutions  are  based  on  the  principle,  illustrated  by  Fig.  57,  that  the  tangent  line  or  tangent  curve 
is  the  limit  of  all  secant  lines  or  curves.  The  problems  on  the  tangencies  of  circles  are  as  follows,  the  number  of  solutions 
in  each  case  being  given :  (1)  To  draw  a  circle  through  three  points.  One  solution.  (2)  Circle  through  two  points  and  tan- 
gent to  a  given  straight  line.  Two  solutions.  (3)  Circle  through  a  given  point  and  tangent  to  two  straight  lines.  Two  solu- 
tions. (4)  Circle  through  two  points  and  tangent  to  a  given  circle.  Two  solutions.  (5)  Circle  through  a  given  point,  tangent 
to  a  given  straight  line  and  a  given  circle.  Four  solutions.  (6)  Circle  through  a  given  point  and  tangent  to  two  given 
circles.  Four  solutions.  (7)  Circle  tangent  to  three  straight  lines,  two  only  of  which  may  be  parallel.  Four  solutions.  (8)  Circle 
tangent  to  two  straight  lines  and  a  given  circle.  Four  solutions.  (Art.  105,  above).  (9)  Circle  tangent  to  two  given  circles  and 
a  given  straight  line.  Eight  solutions.  (10)  Circle  tangent  to  three  giveu  circles.  Eight  eolutions. 


PLANE    PROBLEMS.  — TAPERING    CIRCULAR    ARCS. 


33 


intersection  of  two  lines.  Join  the  given  point  e  with  any  point  /  on  A  B,  and  also  with  some  point 
g  on  CD.  From  any  point  h  on  A  B  draw  h  i  parallel  to  /  g,  then  i  k  parallel  to  g  e,  and  h  k 
parallel  to  /  e.  The  line  k  e  will  fulfill  the  conditions. 

109.  Prob.   24-     To   draw  an  oval  upon   a  given  line.     Describe  a  circle   on  the  given  line,  m  n,  (Fig. 

70)  as  a  diameter.  With  m  and  n  as  centres  describe  arcs,  m  x,  nx, 
radius  m  n.  Draw  m  v  and  n  t  through  v  ''and  t,  the  middle  points 
of  the  quadrants  y  m,  y  n.  Then  m  s  and  r  n  are  the  portions  of 
7?i  x  and  n  x  forming  part  of  the  oval.  Bisect  n  c  at  q  and  draw  q  x. 
Also  bisect  c  q  at  z  and  join  the  latter  with  x.  Bisect  y  b  in  d  and 
draw  /  d  from  /,  the  intersection  of  n  s  and  q  x.  Use  /  as  a  centre, 
and  /  s  as  radius,  for  an  arc  s  k  terminating  on  /  d.  The  intersec- 
tion, h,  of  kfwiih  xz,  is  then  the  next  centre,  and  h  k  the  radius 
of  the  arc  k  I  which  terminates  on.  h  y  produced.  The  oval  is  then 

completed  with   y  as  a   centre  and  radius   y  I.      The  lower  portion   is   symmetrical   with  the   upper,  and 

therefore  similarly  constructed. 

110.  Where   exact  tangency   is   the   requirement,  novices   occasionally   endeavor  to    conceal  a  failure 
to  secure  the   desired   object  by   thickening  the   curve.     Such   a  course  usually   defeats  itself  and  makes 
more  evident  the  error  they  thus   hope  to  conceal.     With  such  instruments  of  precision  as  the  draughts- 
man  employs   there   can   be  but  little   excuse,   if  any,   either  for  overlapping   or  falling  short. 

A   common   error  in   drawing  tangents,  where  the  lines  are  of  apprecia-  Fig-,  n. 

ble  thickness,  is  to  make  the  outsides  of  the  lines  touch ;  whereas  they 
should  have  their  thickness  in  common  at  the  point  of  tangency,  as  at  T 
(Fig.  70),  where,  evidently,  the  centre-lines  a  and  b  of  the  arcs  would  be 
exactly  tangent,  while  the  outer  arc  of  M  would  come  tangent  to  the  inner 
arc  of  N,  and  vice  versa. 

111.  When   either  a  tube   or  a   solid   cylindrical    piece    is    seen    in    the    direction    of   its    axis,  the 
s-ig-.  72.  outline  is,   obviously,   simply   a    circle;    and    often    the    only    way    to    determine 

which   of  the  two   the  circle   represented  would  be  to  notice  which  part   of  said 

^000i^0f^  end  view  was  represented  as  casting  the  shadow.  In  Fig.  72,  if  the  shaded 
arcs  can  cast  shadows,  the  space  inside  the  circles  must  be  open,  and  the  fig- 
ure would  represent  a  portion  of  the  end  view  of  a  boiler  with  its  tubular 
openings. 

OOO^^OO-  ^    exactly    reversing    the    shading    (the    effect    of   which    can    be    seen   by 

turning  the   figure  upside   down)   it  is   converted   into    a    drawing   of   a    number 
of  solid,   cylindrical   pins,  projecting  from   a  plate. 

The  tapering  begins  at  the   extremities   of  a   diameter   drawn   at  45°   to   the   horizontal. 
To  get  a  perfect  taper  on  small  circles  use  the  bow -pen,  and,  after  making   one  complete  circle,  add 
the    extra    thickness    by    a    second   turn,   which    is    to    begin    with    the    pen -point  in  the  air,  the  pen 
being  brought  down  gradually  upon   the   paper,   and  then,   while  turning,   raised   from   it  again. 

On  medium  and  large  circles  the  requisite  taper  can  be  obtained  by  a  different  process,  viz.,  by 
using  the  same  radius  again  but  by  taking  a  second  centre,  distant  from  the  first  by  an  amount  equal 
to  the  proposed  width  of  the  broadest  part  of  the  shaded  arc;  the  line  through  the  two  centres  to 
be  perpendicular  to  that  diameter  which  passes  through  the  extremities  of  the  taper.  The  extra 
thickness  should  be  inside  the  circumference,  not  outside. 


eeoooe 


eeoe 


oeeeeo 


eeo 


34 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


112.    As  exercises  in  concentric  circles  Figs.  73  and  74    will    prove    a   good  test    of  skill.      They 
represent,  either  entire  or  in  section,  a  gymnasium    ring,  the    "annular  torus"  of  mathematical  works. 

F1&.   73.  Fig-  7-^. 


It  is  a  surface  possessing  some  remarkable  properties,  chief  among  which  is  the  fact  that  it  is  the 
only  surface  of  revolution  known  from  which  two  circles  can  he  cut  by  each  plane  in  three  different 
systems  of  planes.*  In  two  of  these  systems  each  plane  will  cut  two  equal  circles  from  the  surface. 

Fig.    75. 


113.  In  Fig.  75  the  same  surface  is  shown  in  front  view,  between  sub -figures  X  and  Y.  The 
axis  of  the  surface  will  be  perpendicular  to  the  paper  at  A.  If  M  N  represents  a  plane  perpendicular 
to  the  paper  and  containing  the  axis,  then  Fig.  X  will  show  the  shape  of  the  cut  or  section.  As 
M  N  was  but  one  of  the  positions  of  a  plane  containing  the  axis,  and  as  the  surface  might  be  gen- 
erated by  rotating  M  N  with  the  circle  a  b  about  the  axis,  it  is  evident  that  in  one  of  the  three 
systems  of  planes  mentioned  in  the  last  article  each  plane  must  contain  the  axis. 

When  a  surface  can  be  generated  by  revolution  about  an  axis  one  of  its  characteristics  is  that 
any  plane  perpendicular  to  the  axis  will  cut  it  in  a  circle.  The  circles  of  Fig.  73  may  then  be,  for  the 
moment,  considered  as  parallel  cuts  by  a  series  of  planes  perpendicular  to  the  axis,  a  few  of  which 


*  Olivier,  Memoires  de  GSomitrie  Descriptive.    Paris,  1851. 


ANNULAR     TORUS.— WARPED    HYPERBOLOID.  35 

may  be  shown  in   m  n,  o  p,   &c.  (Fig.  X ).      Each    of   these  planes    cuts    two    circles  from  the    surface ; 
the   plane  o  p,  for  example,  giving  circles  of   diameters  c  d  and  v  w  respectively. 

A  plane,  perpendicular  to  the  paper  on  the  line  P  Q,  would  be  a  bi-  tangent  plane,  because  tan- 
gent to  the  surface  at  two  points,  P  and  Q;  and  such  plane  would  cut  two  over -lapping  circles  from 
the  torus,  each  W  them  running  partly  on  the  inner  and  partly  on  the  outer  portion  of  the  surface. 
These  sections  are  seen  as  ellipses  in  Fig.  74.  For  the  proof  that  such  sections  are  circles  the  stu- 
dent is  probably  not  prepared  at  this  point,  but  is  referred  to  Olivier's  Seventh  Memoir,  or  to  the 
Appendix.* 

114.  Another  interesting  fact   with    regard    to    the    torus,  is,  that  a   series  of    planes  parallel  to,   but 
not  containing  the  axis,   cut  it   in  a  set   of  curves  called  the  Cassian   ovals   (see   Art.   212),  of   which  the 
Lemniscate  of  Art.   158  is  a  special  case,  and  which  would  result  from  using  a  plane  parallel  to  the 
axis  and  tangent  to  the  surface  at  a  point  on  the  smallest  circle  at  a,   (Fig.  75.)t 

115.  Fig.  Y  is  given  to  illustrate  the  fact  that  from  mere  untapered    outlines,    such    as    compose 
the  central  figure,  we  cannot  determine  the  form  of  the  object.  s^ig.  7s. 

By  shading  eh  /and  DNr  we  get  Fig.  Z,  and  the  form  shown 
in  Fig.  Y  would  be  instantly  recognized  without  the  drawing 
of  the  latter.  An  angular  object  must  therefore  have  shade 
lines,  as  also  the  end  view  of  a  round  object;  but  a  side  view 


of  a  cylindrical   piece  must  either  have  uniform  outlines  or  be   shaded  with   several  lines. 

Thus,  in  Fig.  76,  A  would  represent  an  angular  piece,  while  B  would  indicate  a  circular  cylin- 
der; if  elliptical  its  section  would  be  drawn  at  one  side  as  shown. 

116.  Before  presenting  the  crucial  test  for  the  learner — the  railroad  rail — two  additional  practice 
exercises,  mainly  in  ruling,  are  given  in  Figs.  77  and  78.  The  former  shows  that,  like  the  parabola, 
the  circle  and  hyperbola  can  be  represented  by  their  enveloping  tangents.  The  upper  and  lower 
figures  are  merely  two  views  of  the  surface  called  the  warped  hyperboloid,  from  the  hyperbolas  which 
constitute  the  curved  outlines  seen  in  the  upper  figure.  The  student  can  make  this  surface  in  a 
few  moments  by  stringing  threads  through  equidistant  holes  arranged  in  a  circle  on  two  circular 
discs  of  the  same  or  different  sizes,  but  having  the  same  number  of  holes  in  each  disc.  By  attaching 
weights  to  the  threads  to  keep  them  in  tension  at  all  times,  and  giving  the  upper  disc  a  twist,  the 
surface  will  change  from  cylindrical  or  conical  to  the  hyperboloidal  form  shown. 

Gear  wheels  are  occasionally  constructed,  having  their  teeth  upon  such  a  surface  and  in  the 
direction  of  the  lines  or  elements  forming  it;  but  the  hyperboloid  is  of  more  interest  mathematically 
than  mechanically. 

Begin  the  drawing  by  pencilling  the  three  concentric  circles  of  the  lower  figure.  When  inking, 
omit  the  smaller  circles.  Draw  a  series  of  tangents  to  the  inner  circles,  each  one  beginning  on  the 
middle  circle  and  terminating  on  the  outer.  Assume  any  vertical  height,  t  s',  for  the  upper  figure, 
and  draw  H'  M'  and  P'  R'  as  its  upper  and  lower  limits.  H'  M'  is  the  vertical  projection,  or  eleva- 
tion, of  the  circle  H  K  M  N,  and  all  points  on  the  latter,  as  1,2,3,4,  are  projected,  by  perpendiculars 
to  H'  M',  at  1',  2',  3',  4',  etc.  All  points  on  the  larger  circle  PQR  are  similarly  projected  to  P' R'. 
The  extremities  of  the  same  tangent  are  then  joined  in  the  upper  view,  as  1'  with  1  (a). 


*  An  original  demonstration  by  Mr.  George  F.  Barton  (Princeton,  '95,)  when  a  Junior  in  the  John  C.  Green  School  of 
Science. 

t  These  curves  can  also  be  obtained  by  assuming  two  foci,  as  if  for  ellipse,  but  taking  the  product  of  the  focal  radii  as 
a  constant  quantity,  some  perfect  square.  If  pp'  =  36  then  a  point  on  the  curve  would  be  found  at  the  intersection  of  arcs 
having  the  foci  as  centres,  and  for  radii  2"  and  18",  or  4"  and  9",  etc.  The  Lemniscate  results  when  the  constant  assumed 
is  the  square  of  half  the  distance  between  the  foci. 


36 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


13  1* 


TAPERING    LINES.— RAIL     SECTIONS. 


37 


Part  of  each  line  is  dotted,  to  represent  its  disappearing  upon  an  invisible  portion  of  the  sur- 
face. The  law  of  such  change  on  the  lower  figure  is  evident  from  inspection,  while  on  the  eleva- 
tion the  point  of  division  on  each  line  is  exactly  above  the  point  where  the  other  view  of  the 
same  line  runs  through  H  M  in  the  lower  figure. 

117.  To  reproduce  Fig.  78,  draw  first  the  circle  af  b  n,  then  two  circular  arcs  which  would  con- 
tain a  and  b  if  extended,  and  whose  greatest  distance  from  the  original  circle  is  x,  (arbitrary).  Six- 
teen equidistant  radii  as  at  a,  c,  d,  etc.,  are  next  in  order,  of  which  the  rule  and  45  °  triangle  give 
those  through  a,  d,  f  and  h.  At  their  extremities,  as  m  and  n,  lay  off  the  desired  width,  y,  and  draw, 
toward  the  points  thus  determined,  lines  radiating  from  the  centre.  Terminate  these  last  upon  the 
inner  arcs.  Ink  by  drawing  from  the  centre,  not  through  or  toward  it. 

All  construction  lines  should  be  erased  before  the  tapering  lines  are  filled  in.  The  "filling  in" 
may  be  done  very  rapidly  by  ruling  the  edges  in  fine  lines  at  first,  then  opening  the  pen  slightly 
and  beginning  again  where  the  opening  between  the  lines  is  apparent  and  ruling  from  there,  adding 
thickness  to  each  edge  on  its  inner  side.  It  will  then  be  but  a  moment's  work  to  fill  in,  free- 
hand, with  the  Falcon  pen  or  a  fine -pointed  sable -brush,  between  the — now  heavy — edge-lines  of  the 

taper.  To  have  the  pen  make  a  coarse  line 
when  starting  from  the  centre  would  destroy 
the  effect  desired. 

118.  The  draughtsman's  ability  can  scarcely 
be  put  to  a  severer  test  on  mere  outline  work 
than  in  the  drawing  of  a  railroad  rail,  so  many 
are  the  changes  of  radii  involved. 

As  previously  stated,  where  tangencies  to 
straight  lines  are  required,  the  arcs  are  to  be 
drawn  first,  then  the  tangents. 

Figs.  79  and  80  are  photo -engravings  of 
rail  sections,  showing  two  kinds  of  "finish." 
Fig.  80  is  a  "working  drawing"  of  a  Penn- 
sylvania Railroad  rail,  scale  7  :  8.  This  makes 
one  of  the  handsomest  plates  that  can  be 
undertaken,  if  finished  with  shade  lines,  as  in 
Fig.  79,  section  -  lined  with  Prussian  blue,  and 
the  dimension  lines  drawn  in  carmine. 

A   still  higher  effect  is  shown  in  the  wood- 
cut  on  page  85,   the  rail  being  represented  in   oblique   projection   and  shaded. 

Begin  Fig.  80  by  drawing  the  vertical  centre-line,  it  being  an  axis  of  symmetry.  Upon  it  lay 
off  5"  for  the  total  height,  and  locate  two  points  between  the  top  and  base,  at  distances  from  them 
of  If"  and  -J"  respectively;  these  to  be  the  points  of  convergence  of  the  lower  lines  of  the  head  and 
sloping  sides  of  the  base.  From  these  points  draw  lines,  at  first  indefinite  in  length,  and  inclined 
13°  to.  the  horizontal.  The  top  of  the  head  is  an  arc  of  10"  radius,  subtended  by  an  angle  of  9°. 
This  changes  into  an  arc  of  ^"  radius  on  the  upper  corner,  with  its  centre  on  the  side  of  said  9° 
angle.  The  sides  of  the  head  are  straight  lines,  drawn  at  4°  to  the  vertical,  and  tangent  to  the 
corner  arcs.  The  thin  vertical  portion  of  the  rail  is  called  the  web,  and  is  %%"  wide  at  its  centre. 
The  outlines  of  the  web  are  arcs  of  8"  radius,  subtended  by  angles  of  15°,  centres  on  line  marked 
"centre  line  of  bolt  holes," 


E   '  r 


38 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


The  weight  per  yard  of  the  rail  shown  is  given  as  eighty- 
five  pounds,*  from  which  we  know  the  area  of  the  cross -section 
to  be  eight  and  one -half  square  inches,  since  a  bar  of  iron  a 
yard  long  and  one  square  inch  in  cross  -  section  weighs,  approx- 
imately, ten  pounds.  (10.2  Ibs.,  average). 

The  proportions  given  are  slightly  different  from  those 
recommended  in  the  report+  of  the  committee  appointed  by  the 
American  Society  of  Civil  Engineers  to  examine  into  the  proper 
relations  to  each  other  of  the  sections  of  railway  wheels  and 
rails.  There  was  quite  general  agreement  as  to  the  following 
recommendations :  a  top  radius  of  twelve  inches ;  a  quarter-inch 
corner  radius;  vertical  sides  to  the  web;  a  lower -corner  of 
one-sixteenth  inch,  and  a  broad  head  relatively  to  the  depth. 


eo. 


•See  the  Appendix  for  dimensions  of  a  100-lb.  rail. 


t  Transactions  A.  S.   C.   E.,  January,  1891. 


EXERCISES    FOR     THE    IRREGULAR    CURVE. 


39 


CHAPTER     F. 

THE  HELIX.  — CONIC  SECTIONS.— HOMOLOGICAL  PLANE  CURVES  AND  SPACE-FIGURES.  — LINK- 
MOTION  CURVES.— CENTROIDS.  — THE  CYCLOID.  — COMPANION  TO  THE  CYCLOID.  — THE  CUR- 
TATE TROCHOID.  — THE  PROLATE  TROCHOID.  — HYPO-,  EPI-,  AND  PERI-TROCHOIDS.  — SPECIAL 
TROCHOIDS— ELLIPSE,  STRAIGHT  LINE,  LIMACON,  CARDIOID,  TRISECTRIX,  INVOLUTE,  SPIRAL 
OF  ARCHIMEDES.  —  PARALLEL  CU  RVES.  —  CONCHOID.  —  QU  ADRATRIX.  —  CISSOID.  —  TRACTRIX.  — 
WITCH  OF  AGNESI.  — CARTESIAN  OVALS.  — CASSIAN  OVALS. —CATENARY.  — LOGARITHMIC 
SPIRAL.— HYPERBOLIC  SPIRAL.— THE  LITUUS.— THE  IONIC  VOLUTE. 


119.  There  are  many  curves  which  the  draughtsman    has    frequent    occasion  to  make,  whose  con- 
struction involves   the   use   of  the  irregular  curve.     The   more  important  of  these   are  the   Helix ;    Conic 
Sections  —  Ellipse,  Parabola   and  Hyperbola ;    Link-motion   curves  or  point -paths;    Centroids;    Trochoids; 
the  Involute  and  the  Spiral  of  Archimedes.      Of  less    practical    importance,  though  equally  interesting 
geometrically,  are  the  other  curves  mentioned  in  the  heading. 

The  student  should  become  thoroughly  acquainted  with  the  more  important  geometrical  properties 
of  these  curves,  both  to  facilitate  their  construction  under  the  varying  conditions  that  may  arise  and 
also  as  a  matter  of  education.  Considerable  space  is  therefore  allotted  to  them  here. 

At  this  point  Art.  .58  should  be  reviewed,  and  in  addition  to  its  suggestions  the  student  is  fur- 
ther advised  to  work,  at  first,  on  as  large  a  scale  as  possible,  not  undertaking  small  curves  of  sharp 
curvature  until  after  acquiring  some  facility  with  the  curved  ruler. 

THE     HELIX. 

120.  The  ordinary  helix  is  a  curve  which  cuts  all  the  elements  of  a  right  cylinder  at  the  same 
angle.      Or  we  may  define  it  as  the  curve  which  would    be  generated    by   a  point    having  a  uniform 
motion  around  a  straight  line,  combined  with  a  uniform  motion  parallel  to  the  line. 


fflUUUUUUUUUUUUUL 

The  student  can  readily  make  a  model  of  the  cylinder  and  helix  by 
drawing  on  thick  paper  or  Bristol-board  a  rectangle  A"  B"  C" D"  (Fig.  81) 
and  its  diagonal,  D" B";  also  equidistant  elements,  as  m"b",  n"c",  etc. 
Allow  at  the  right  and  bottom  about  a  quarter  of  an  inch  extra  for  over- 
lapping, as  shown  by  the  lines  x  y  and  a  z.  Cut  out  the  rectangle  z  x ;  also  cut  a  series  of  vertical 
slits  between  D"  C"  and  z.s;  put  mucilage  between  B"  C"  and  xy;  then  roll  the  paper  up  into 
cylindrical  form,  bringing  A"  D"  t"  h"  in  front  of  and  upon  the  gummed  portion,  so  that  A"  D" 


40  THEORETICAL    AND    PRACTICAL    GRAPHICS. 

will  coincide  with  B"  C".  The  diagonal  D"  B"  will  then  be  a  helix  on  the  outside  of  the  cylinder, 
but  half  of  which  is  visible  in  front  view,  as  D'T,  (see  right-hand  figure);  the  other  half,  T A', 
being  indicated  as  unseen. 

To  give  the  cylinder  permanent  form  it  can  then  be  pasted  to  a  cardboard  base  by  mucilage  on 
the  under  side  of  the  marginal  flaps  below  D"  C",  turning  them  outward,  not  in  toward  the  axis. 

The  rectangle  A" B"  C" D"  is  called  the  development  of  the  cylinder;  and  any  surface  like  a 
cylinder  or  cone,  which  can  be  rolled  out  on  a  plane  surface  and  its  equivalent  area  obtained  by 
bringing  consecutive  elements  into  the  same  plane,  is  called  a  developable  surface.  The  elements  m"b", 
n"c",  etc.,  of  the  development  stand  vertically  at  b,  c,  d  .  .  .  .  g  of  the  half  plan,  and  are  seen  in  the 
elevation  at  m'b',  n'c',  o' d',  etc.  The  point  3',  where  any  element,  as  c',  cuts  the  helix,  is  evidently 
as  high  as  3",  where  the  same  point  appears  on  the  development.  We  may  therefore  get  the  curve 
D' T A'  by  erecting  verticals  from  b,c,d....g,  to  meet  horizontals  from  the  points  where  the  diago- 
nal D"  B"  crosses  those  elements  on  the  development.  D"  C"  obviously  equals  2  irr,  where  r=OD. 

The  shortest  method  of  drawing  a  helix  is  to  divide  its  plan  (a  circle)  and  its  pitch  (D' A',  the  rise 
in  one  turn)  into  the  same  number  of  equal  parts;  then  verticals  bm',  en',  etc.,  from  the  points  of 
division  on  the  plan,  will  meet  the  horizontals  dividing  the  pitch,  in  points  2',  3',  etc.,  of  the  desired 
curve. 

The  construction  of  the  helix  is  involved  in  the  designing  of  screws  and  screw-propellers,  and  in 
the  building  of  winding  stairs  and  skew-arches. 

Mathematically,  both  the  curve  and  its  orthographic  projection  are  well  worth  study,  the  latter 
being  always  a  sinusoid,  and  becoming  the  companion  to  the  cycloid  for  a  45°-helix.  (Arts.  170  and  171). 

For  the  conical  helix,  seen  in  projection  and  development  as  a  Spiral  of  Archimedes,  see  Art.  191. 

THE     CONIC     SECTIONS. 

121.  'The  ellipse,  parabola  and    hyperbola  are  called  conic  sections  or  conies    because  they    may  be 
obtained  by  cutting  a  cone  by  a  plane.      We  will,  however,  first  obtain  them  by  other  methods. 

According  to  the  definition  given  by  Boscovich,  the  ellipse,  parabola  and  hyperbola  are  curves  in 
which  there  is  a  constant  ratio  between  the  distances  of  points  on  the  curve  from  a  certain  fixed 
point  (the  focus)  and  their  distances  from  a  fixed  straight  line  (the  directrix). 

Referring  to  the  parabola,  Fig.  82,  if  S  and  B  are  points  of  the  curve,  F  the  focus  and  XY  the 
directrix,  then,  if  SF:ST::BF:BX,  we  conclude  that  B  and  S  are  points  of  a  conic  section. 

122.  The    actual    value    of   such    ratio    (or    eccentricity')    may    be    1,  or    either   greater  or  less  than 
unity.      When   SF  equals  S  T   the    ratio    equals    1,  and    the   relation    is    that    of   equality,    or    parity, 
which   suggests  the   parabola. 

123.  If  it  is  farther  from  a  point    of  the    conic    to    the    focus  than  to  the  directrix  the  ratio  is 
greater  than  1,   and  the  hyperbola  is  indicated. 

124.  The  ellipse,  of  course,  comes  in  for  the  third  possibility    as    to  ratio,  viz.,  less  than  1.      Its 
construction  by  this  principle  is  not  shown  in  Fig.  82  but  later,  (Art.  142),  the  method   of  generation 
here  given  illustrating  the  practical  way  in  which,  in  landscape  gardening,  an  elliptical  plat  would  be 
laid  out;  it  is  therefore,  called  the  construction  as  the  "gardener's  ellipse." 

Taking  A  C  and  D  E  as  representing  the  extreme  length  and  width,  the  points  F  and  Fl  (foci) 
would  be  found  by  cutting  A  C  by  an  arc  of  radius  equal  to  one -half  A  C,  centre  D.  Pegs  or  pins 
at  F  and  F^ ,  and  a  string,  of  length  A  C,  with  ends  fastened  at  the  foci,  complete  the  preliminaries. 
The  curve  is  then  traced  on  the  ground  by  sliding  a  pointed  stake  against  the  string,  as  at  P,  so 
that  at  all  times  the  parts  Ft  P,  F  P,  are  kept  straight. 


CONIC    SECTIONS. 


41 


125.  According  to  the  foregoing  construction  the  ellipse  may  be  defined  as  a  curve  in  which  the 
sum  of  the  distances  from  any  point  of  the  curve  to  two  fixed  points  is  constant.     That   constant  is  evidently 
the   longer   or    transverse   (major)   axis,    A  C.      The    shorter    or    conjugate  (or  minor')   axis,   D  E,  is   perpen- 
dicular to  the   other. 

With  the  compasses  we  can  determine  P  and  other  points  of  the  ellipse,  by  using  F  and  F^  as 
centres,  and  for  radii  any  two  segments  of  A  C.  Q,  for  example,  gives  A  'Q  and  C  Q  as  segments. 
Then  arcs  from  F  and  Ft ,  with  radius  equal  to  Q  C,  would  intersect  arcs  from  the  same  centres, 
radius  QA,  in  four  points  of  the  ellipse,  one  of  which  is  P. 

126.  By   the    Boscovich    definition   we    are    also    enabled    to    construct    the    parabola   and   hyperbola 
by   continuous   motion   along  a   string. 

For  the  parabola  place  a  triangle  as  in  Fig.  82,  with  its  altitude  GX  toward  the  focus.  If  a 
string  of  length  G  X  be  fastened  at  G,  stretched  tight  from  G  to  any  point  B,  by  putting  a  pencil 
at  B,  then  the  remainder  B  X  swung  around  and  the  end  fastened  at  F,  it  is  then,  evidently,  as 
far  from  B  to  F  as  it  is  from  B  to  the  directrix;  and  that  relation  will  remain  constant  as  the  tri- 
angle is  slid  along  the  directrix,  if  the  pencil  point  remains  against  the  edge  of  the  triangle  so  that 
the  portion  of  the  string  from  G  to  the  pencil  is  kept  straight. 


127.  For    the    hyperbola,   (Fig.  82),  the    construction    is    identical    with    the    preceding,    except    that 
the   string   fastened   at  /  runs   down  the   hypothenuse,   and   equals   it  in   length. 

128.  Referring   back   to   Fig.   35,  it   will   be   noticed   that   the    focus  and  directrix  of  the  parabola 
are    there    omitted;    but    the    former    would    be    the    point   of   intersection    of   a    perpendicular  from   A 
upon   the  line  joining  C  with  E.     A   line  through  A,   parallel  to   C  E,   would   be  the   directrix. 

129.  Like  the    ellipse,  the    hyperbola   can   be   constructed   by  using   two    foci,  but    whereas  in  the 
ellipse    (Fig.    82)    it    was    the    sum    of   two    focal    radii    that    was    constant,    i.e.,   FP+F1P^=FD  + 


42 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


FtD  =  A  C  (the  transverse  axis),  it  is  the  difference  of  the  radii 
that  is  constant  for  the  hyperbola. 

In  Fig.  83  let  A  B  be  the  transverse  axis  of  the  two 
arcs,  or  "branches,1"  which  make  the  complete  hyperbola;  then 
using  p  and  p  to  represent  any  two  focal  radii,  as  FQ  and 
FI  Q,  or  F R  and  F^  R,  we  will  have  p — p'=A  B,  the  constant 
quantity. 

To  get  a  point  of  the  curve  in  accordance  with  this  prin- 
ciple we  may  lay  off  from  either  focus,  as  F,  any  distance 
greater  than  FB,  as  FJ,  and  with  it  as  a  radius,  and  F  as 
a  centre,  describe  the  indefinite  arc  JR.  Subtracting  the  con- 
stant, A  B,  from  FJ,  by  making  JE=AB,  we  use  the 
remainder,  FE,  as  a  radius,  and  Ft  as  a  centre,  to  cut  the 
first  arc  at  R.  The  same  radii  will  evidently  determine  three 
other  points  fulfilling  the  conditions. 


-.  03. 


130.    The  tangent  to  an  hyperbola  at  any  point,  as   Q,  bisects    the   angle  FQFlt  between   the    focal 


radii. 

In  the  ellipse,  (Fig.  82),  it  is  the  external  angle  between  the  radii  that  is  bisected  by  the  tan- 
gent. 

In  the  parabola,  (Fig.  82),  the  same  principle  applies,  but  as  one  focus  is  supposed  to  be  at 
infinity,  the  focal  radius,  B  G,  toward  the  latter,  from  any  point,  as  B,  would  be  parallel  to  the  axis. 
The  tangent  at  B  would  therefore  bisect  the  angle  FBX. 

131.  The   ellipse   as   a   circle   viewed  obliquely.    If  ARMBF  (Fig.   84)   were  a  circular  disc  and  we 

'- e'4-  were  to  rotate  it  on  the  diameter  A  B,  it  would  become 

narrower  in  the  direction  FE  until,  if  sufficiently  turned, 
only  an  edge  view  of  the  disc  would  be  obtained.  The 
axis  of  rotation  A  >B  would,  however,  still  appear  of  its 
original  length.  In  the  rotation  supposed,  all  points 
not  on  the  axis  would  describe  circles  about  it  with 
their  planes  perpendicular  to  it.  M,  for  example, 
.would  describe  an  arc,  part  of  which  is  shown  in 
MMj,  which  is  straight,  as  the  plane  of  the  arc  is 
seen  "edge-wise."  If  instead  of  a  circular  disc  we  turn 
an  elliptical  one,  A  C  B  D,  upon  its  shorter  axis  CD,  it 
is  obvious  that  B  would  apparently  approach  0  on  one 
side  while  A  advanced  on  the  other,  and  that  the  disc 
could  reach  a  position  in  which  it  would  be  projected  in 
the  small  circle  CkD.  If,  then,  the  axes  of  an  ellipse  are  given,  as  A  B  and  CD,  use  them  as  diam- 
eters of  concentric  circles;  from  their  centre,  0,  draw  random  radii,  as  0  T,  OK;  then  either,  as  0  T, 
will  cut  the  circles  in  points,  t  and  T,  through  which  a  parallel  and  perpendicular,  respectively,  to 
the  longer  axis,  will  give  a  point  T1  of  an  ellipse. 

The  relation  just  illustrated  is  established  analytically  in  the  Appendix. 

132.  If  TS   is    a    tangent   at    T  to    the  large   circle,    then   when    T  has    rotated    to    T,    we    shall 
have  T,  S  as  a  tangent  to  the  ellipse  at  the  point   derived   from  T,  the  point  S  having  remained  con- 
stant,  being  on  the  axis   of  rotation. 


CONIC    SECTIONS. 


43 


as. 


Similarly,   if  a   tangent  at   Rl   were   wanted,  we  would   first   find   r,  corresponding  to   R,;   draw   the 
tangent  rJ  to   the   small   circle;    then  join   Rl  to  J,   the   latter   on   the   axis   and   therefore   constant. 

133.  Occasionally   we   have   given   the   length   and   inclination   of  a   pair   of  diameters   of  the  ellipse, 

making  oblique  angles  with  each  other.  Such  diameters, 
are  called  conjugate,  and  the  curve  may  be  constructed 
upon  them  thus:  Draw  the  axies  TD  and  H  K  at  the 
assigned  angle  D  0  H;  construct  the  parallelogram  MN 
X  Y;  divide  D  M  and  D  0  into  the  same  number  of 
equal  parts  J  then  from  K  draw  lines  through  the  points 
of  division  on  D  0,  to  meet  similar  lines  drawn  through  H 
and  the  divisions  on  D  M.  The  intersection  of  like -num- 
bered lines  will  give  points  of  the  ellipse. 

134.  It  is   the   law   of  expansion   of  a   perfect  gas    that    the    volume    is    inversely   as    the    pressure. 
That  is,  if  the   volume  be   doubled   the   pressure   drops   one -half;    if  trebled   the   pressure   becomes   one- 
third,   etc.     Steam,  not  being   a   perfect   gas,  departs   somewhat   from  the   above   law,  but   the   curve   indi- 
cating  the   fall   in   pressure   due   to   its   expansion   is   compared   with   that  for  a   perfect  gas. 

To  construct  the  curve  for  the  latter  let  us  suppose  C  L  K  G  (Fig.  86)  to  be  a  cylinder  with  a 
volume  of  gas  C  G  b  c  behind  the  piston.  Let  c  b  indicate  the 
pressure  before  expansion  begins.  If  the  piston  be  forced  ahead 
by  the  expanding  gas  until  the  volume  is  doubled,  the  pressure 
will  drop,  by  Boyle's  law,  to  one -half,  and  will  be  indicated  by 
td.  For  three  volumes  the  pressure  becomes  v  f,  etc.  The  curve 
csx  is  an  hyperbola,  -of  the  form  called  equilateral,  or  rectangular. 

Suppose  the  cylinder  were  infinite  in  length.  Since  we  cannot 
conceive  a  volume  so  great  that  it  could  not  be  doubled,  or  a  pressure  so  small  that  it  could  not 
be  halved,  it  is  evident  that  theoretically  the  curve  c  x  and  the  line  G  K  will  forever  approach  each 
other  yet  never  meet;  that  is,  they  will  be  tangent  at  infinity.  In  such  a  case  the  straight  line  is 
called  an  asymptote  to  the  curve. 

135.  Although    the   right    cone    (i.  e.,   one   having    its    axis    perpendicular    to    the   plane    of  its    base) 
is   usually   employed   in   obtaining   the   ellipse,   hyperbola   and   parabola,   yet   the    same   kind    of   sections 

can  be  cut  from  an  oblique  or  scalene  cone  of  circular  base,  as  V.  A  B, 
Fig.  87.  Two  sets  of  circular  sections  can  also  be  cut  from  such  a 
cone,  one  set,  obviously,  by  planes  parallel  to  the  base,  while  the 
other  would  be  by  planes  like  CD,  making  the  same  angle  with  the 
lowest  element,  V B,  that  the  highest  element,  V A,  makes  with  the 
base.  The  latter  sections  are  called  sub -.contrary.  Their  planes  are 
perpendicular  to  the  plane  V A  B  containing  the  highest  and  lowest 
elements — principal  plane,  as  it  is  termed. 

To  prove  that  the  sub  -  contrary  section  x  y  is  a  circle,  we  note 
that  both  it  and  the  section  m  n — the  latter  'known  to  be  a  circle 
because  parallel  to  the  base  —  intersect  in  a  line  perpendicular  to  the 
paper  at  o.  This  line  pierces  the  front  surface  of  the  cone  at  a  point  we  may  call  r. 
It  would  be  seen  as  the  ordinate  o  r  (Fig.  88),  were  the  front  half  of  the  circle  m  n 
rotated  until  parallel  to  the  paper.  Then  or^  —  omxon.  But  in  Fig.  87  we  have 
om:oy::or:o  n,  whence  oy  X  ox  =  omx  on  —  or"1,  proving  the  section  x  y  circular. 


as. 
o 


44 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


M 


Were  the  vertex  of  a  scalene  cone  removed  to  infinity,  the  cone  would  become  an  oblique  cylinder 
with  circular  base;   but  the  latter  would  possess  the  property  just  estab- 
lished for  the  former. 

136.  The  most  interesting  practical  application  of  the  sub -contrary 
section  is  in  Stereographic  Projection,  one  of  the  methods  of  represent- 
ing the  earth's  surface  'on  a  map.  The  especial  convenience  of  this 
projection  is  due  to  the  fact  that  in  it  every  circle  is  projected  as  a 
circle.  This  results  from  the  relative  position  of  the  eye  (or  centre  of 
projection)  and  the  plane  of  projection;  the  latter  is  that  of  some 


XS:G  R::JS:J  R 

^j  F; 

.'.  Dr;:DZ(=GR)::j  F;:J  R 
J  f^>J  R 

.-.  curve  is  an  hyperbola. 


<JF=QB=OA 

=  B  W=AN  = 

••  curve  M  Q  Mtis  a 

parabola. 


-.  ©O. 


great  circle  of  the  earth,  and  the  centre  of  projection  is  located  at  the  pole  of  such   circle. 


CONIC    SECTIONS. 


45 


137.  In  Fig.   89  let  the  circle  ABE  represent   the    equator;    MN  the    plane    of  a    meridian,  also 
taken  as  the  plane   of  projection;    AB  any  circle  of  the  sphere;   E  the  position  of  the  eye:   then  a  b, 
the  projection  of  A  B  on  plane  MN,  is  a  circle,  being  a  sub -contrary  section  of  the  cone  E.AB. 

138.  We  now  take  up  the  conic  sections  as  derived  from  a  right  cone. 

A  complete  cone  (Fig.  90)  lies  as  much  above  as  below  the  vertex.  To  use  the  term  adopted 
from  the  French,  it  has  two  nappes. 

Aside  from  the  extreme  cases  of  perpendicularity  to  or  containing  the  axis,  the  inclination  of  a  plane 
cutting  the  cone  may  be 

(a)  Equal    to    that    of   the    elements    (see  the  remark  in  Art.  4),  therefore  parallel  to  one  element, 
giving    the    parabola,   as   M  Q  Ml    (Fig.   90) ;    the    plane   k  q  M   being    parallel    to    the    element    V  U,  and 
therefore  making  with  the  base  the  same  angle,   6,  as  the  latter. 

(b)  Greater   than    that   of   the   elements,   causing  the   plane  to   cut   both   nappes,  and  giving  a  two- 
branch   curve,   the   hyperbola,  as   DJE  and  fhg   (Fig.    90). 

(c)  Less  than  that  of  the  elements,  the  plane  therefore   cutting  all  the  elements  on  one  side  of  the 
vertex,  giving  a  closed  curve,  the  ellipse;  as  KsH,  Fig.  91. 

139.  Figures  90  and  91,   with   No.  4  of  Plate   2,  are 
not   only  stimulating   examples   for  the   draughtsman,  but 
they    illustrate    probably    the    most    interesting    fact    met 
with    in    the    geometrical    treatment    of    conies,    viz.,   that 
the  spheres  which  are  tangent  simultaneously  to  the  cone 
and  the  cutting  planes,  touch  the  latter  in  the  foci  of  the 
conies;    while  in   each   case  the   directrix   of  the  curve  is 
the    line    of    intersection    of   the    cutting    plane    and    the 
plane   of  the   circle   of  tangency   of  cone   and   sphere. 

To  establish  this  we  need  only  employ  the  well- 
known  principles  that  (a)  all  tangents  from  a  point  to 
a  sphere  are  equal  in  length,  and  (b)  all  tangents  are 
equal  that  are  drawn  to  a  sphere  from  points  equidis- 
tant from  its  centre.  In  both  figures  all  points  of  the 
cone's  bases  are  evidently  equidistant  from  the  centres 
of  the  tangent  spheres. 

140.  On   the   upper   nappe   (Fig.  90)   let   SH  be  the 
circle    of   contact    of   a    sphere    which    is    tangent    at    F: 
PHl  of  the  circle    cuts    the    plane    of   section  in    Pm. 


v 

cutting    plane    P  L  K.       The    plane 
any    point    of   the    curve    DJE,    J 


to    the 

If  D   is 

another  point,  and  we  can  prove  the  ratio  constant  (and  greater  than  unity)  between  the  distances 
of  D  and  J  from  Fl  and  their  distances  to  P m,  then  the  curve  DJE  must  be  an  hyperbola,  by  the 
Boscovich  definition;  Fl  must  be  the  focus  and  Pm  the  directrix. 

D  Fl  is  a  tangent  whose  real  length  is  seen  at  XS.  JFl  and  JS  are  equal,  being  tangents  to 
the  sphere  from  the  same  point.  We  have  then  the  proportion  XS:  G  R: :  JS:  J  R,  or  DFl:DZ:: 
JF1 :  JR.  Since  JS  and  its  equal  J  Fl  are  greater  than  J  R,  and  the  ratio  J'Fl  to  J  R  is  constant, 
the  proposition  is  established. 

141.  For  the  parabola  on  the  lower  nappe,  since  the  plane  Mqk  is  inclined  at  the  angle  6 
made  by  the  elements,  we  have  QA=QB  (opposite  equal  angles),  and  Q£  equal  QF  (equal  tan- 
gents). MF=BW=Mo,  therefore  MF:  Mo: :  QF:  QB(—  QA),  and  it  is  as  far  from  M  to  the  focus 
F  as  to  the  directrix  ax,  fulfilling  the  condition  essential  for  the  parabola. 


46 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


142.  For  the  ellipse  KsH,   (Fig.   91),  we   have  PX  and  NT  as    the    lines    to    be    proven    direc- 
trices, and  F  and  Ft  the   points   of  tangency  of  two  spheres.      Let  s  be  any   point   of  the  curve   under 
consideration,  and    V L  the  element   containing   s.      This   element   cuts  the  contact  circles  of  the   spheres 
in   a  and   A.      A   plane   through  the   cone's   axis   and   parallel  to   the    paper  would   contain   ot,   ov  and 
vn.      Prolong  vn  to   meet   a  line    VR  that  is   parallel   to  K H.      Join   R  with   a,  producing   it  to   meet 
PX  at  r.      In   the   triangles   asr  and  aVR  we  have  s  a ,:  sr :  :Va:VR.      But  sa  —  sF  (equal   tangents) 
and    similarly     Va=Vn;    hence   s  F:  sr :  :Vn :  VR,   which    ratio    is    less    than    unity;     therefore    s    is    a 
point   on   an   ellipse.* 

The  plane  of  the  intersecting  lines  Va  and  R  r  cuts  the  plane  M N  in  A  T,  which  is  therefore 
parallel  to  or;  hence  sA  :  s  T:  :Vn  :V  R.  But  sA  =  sFl;  therefore  s  Fl :  s  T:  :Vn  :  VR,  the  same  ratio 
as  before. 

143.  If    the    plane    of    section    P  N    were    to     approach    parallelism    to     V  C,  the    point    R    would 
advance  toward  n,  and  when   VR    became   Vn    the    plane    would    have    reached    the    position    to    give 
the   parabola. 


*  Schlomilch,   Geometrie  des  Maasses,    1874. 


COKIC    SECTIONS    AS    HOMOLOGOUS    FIGURES. 


47 


144.  The   proof   that   KsH   (Fig.  91)    is   an   ellipse   when,  the   curve   is   referred   to    tivo   foci    is    as 
follows:     K  F=  Km;    K  F,  =  Kt;    therefore    K  F  +  K  Fl  =  Km  +  Kt  =  tm  =  xn  =.  2  K  F  +  F  F,  = 
2HF+FF,;   i.e.,   K  F  =  H  F,. 

Since  s  F  =  s  a,  and  s  f\  —  s  A,  we  have  s  F  +  sF1  =  sa  +  sA  —  Aa  =  tm  =  xn=HK.  The 
sum  of  the  distances  from  any  point  s  to  the  two  fixed  points  F  and  Fl  is  therefore  constant,  and 
equal  to  the  longer  axis,  H  K. 

HOMOLOGOUS     PLANE     AND     SPACE     FIGURES. 

145.  Before  leaving   the   conic    sections,  their   construction   will    be    given    by   the   methods   of    Pro- 
jective   Geometry.      (At   this   point   review   Arts.   4   and   9). 

In    Fig.   92,   if  Ss   is   a   centre    of   projection,   then    the    figure   A1  Bl  C1    is    the    central   projection  of 

AlBl  C, .     The   points   A '   and   A ,   are   corresponding  points,   being   in   different   pianos   but   collinear   with 

6',.      Similarly   Bl   corresponds   to   Bl,   and    C"   to    (7,. 

With   S,   as   the   centre   of  projection   we   have   the   figure   A.2B2C.^   corresponding   to   A1  Bl  C1. 

We    are    to    show    that    some    point    0    can    be    found,    in    the   plane    of   the    figures   A1B1C1    and 

A^B,C2,   to   which — and   to   each   other — those  figures  bear  the   same   relation  as  that  existing   between 

each   of  them   and   A1  B1  C1   when   considered   in   connection   with   one   of  the  $- centres.      This   compels 

the   points  of  intersection  of  corresponding   lines  to   be   collinear.      Figures  standing   in  these   relations   to 

a  point  and   a   line  in   their   plane   are   called  homologous.     The  point  is  called   a  centre  of  homology;  the 

line  an  axis  of  homology.     Points  collinear  with   0,  as  A  l  and  A  2 ,  are  homologous  points.       T^ig-.  ©3. 
To   illustrate   these   statements,  join   S.f   with   5,   and   prolong  to   0, 

to    meet    the    plane    of    the    figures   A1B1C1   and   A.^B.^C^.      Using 

the   technical   term   trace  for    the    intersection    of   a    line 

plane   or   of  one   plane   with   another,   we   see   that  as   th 

AjA2  is   the  horizontal  trace   of  the   plane   determined 

the    lines    joining   A1   with   S^   and   St,  it    must    contain 

the   horizontal   trace,    0,   of   the   line    joining  S.2  with 

S,.      But    this    puts    A 2    and    Al    into    the    same 

relation    with    0    that    A.^    and    A1    sustain    to 

S2;     or  that   of  A1   and   Al   to   £,. 

Again,   A1  c0    is   the   trace,   on    the   vertical 

plane,  of  any  plane  containing  A1  B\  This 
plane  cuts  the  "axis  of  homology,"  tlm,  in 
c0.  As  A1Bl  lies  in  the  plane  of  Sl  and 
A1  Bl,  and  in  the  horizontal  plane  as  well,  it 
can  only  meet  the  vertical  plane  in  c0,  the 
point  of  intersection  of  all  these  planes.  Sim- 
ilarly we  find  that  A1  C1  and  AjC\,  if  pro- 
longed, meet  the  axis  at  the  point  b0;  cor- 
respondingly Bl  C"  and  Bl  C,  meet  at  «0. 
But  A1  B1  and  A2B.,,  being  corresponding 
lines,  lie  in  the  plane  with  <S2,  though  belonging  to  figures  in  two  other  planes;  they  must,  there- 
fore, meet  also  at  the  same  point,  c0;  and  similarly  for  the  other  lines  in  the  figures  used  with  <S'2. 
146.  Were  A1B1C\  a  circle,  and  all  its  points  joined  with  Sn  the  figure  A1  B1  C1  would  obviously 
be  an  ellipse;  equally  so  were  A2B.2C2  a  circle  used  in  connection  with  St.  We  may,  therefore, 


48 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


substitute  a  circle  for  A1B.tC1,  and  using  0  on  the  same  plane  with  it  get  an  ellipse  in  place  of 
the  triangle  A ,  B  l  Cl .  Before  illustrating  this,  it  is  necessary  to  show  the  relation  of  the  axis  to  the 
other  elements  of  the  problem,  and  supply  a  test  as  to  the  nature  of  the  conic. 

147.  First  as  to   the   axw,   and  employing   again   for  a  time   a   space   figure   (Fig.  93),  it   is   evident 
that    raising    or    lowering    the    horizontal    plane    c  X  Y  parallel    to    itself,   and    with    it,    necessarily,  the 
axis,   would   not  alter  the  kind  of  curve  that  it  would  cut   from  the   cone   S.  H  A  B,  were   the   elements 
of  the  latter   prolonged.      But  raising   or  lowering  the   centre  S,  while  the  base  circle  AHBt  remained, 
as  before,   in   the   same  place,  would   decidedly   affect   the   curve.      Where   it  is,  there  are  two   elements 
of  the   cone,  SA   and  SB,   which   would   never    meet    the    plane   cXY.      The    shaded    plane    containing 
those  elements  meets  the   vertical   plane  in   "vanishing   line   (a),"   parallel   to    the    axis.      This    contains 
the    projections,  A    and  B,   of   the    points    at    infinity   where    the    lower    plane    may    be    considered    as 
cutting  the   elements   SA   and   SB.      Were  S  and   the   shaded    plane   raised  to   the   level   of  H,  making 
"vanishing  line  (a)"  tangent  to  the  base,  there  would  be  one  element,  S  H,  of  the  cone,  parallel  to  the 
lower  plane,   and  the    section   of  the   cone   by   the  latter  would  be  the  parabola;    as  it  is,  the   hyperbola 
is  indicated.      The    former    would   have  but   one   point  at  infinity;    the  latter,   two. 

148.  Raise   the   centre  S  so   that  the   vanishing   line   does  not   cut  the  base,   and   evidently   no  line 
from  S  to   the  base  would   be   parallel  to   the  lower   plane;    but  the  latter  would   cut  all   the   elements 
on   one   side   of  the   vertex,   giving  the   ellipse. 

149.  Bearing  in  mind  that  the  projection   of  the   circle  AHBt  is   on   the  lower   plane   produced, 
if  we  wish  to  bring  both  these  figures  and  the  centre  S  into  one  plane  without  destroying  the  relation 
between  them,  we  may  imagine  the  end  plane  QLX  removed,  the  rotation  of   the  remaining  system 


occurring  about  crl  in  a  manner  exactly  similar  to   that   which  would    occur   were  iojc  a  system  of 
four  pivoted  links,  and  the  point  o  pressed  toward  c.      The  motion  of  S  would  be  parallel  and  equal 


CONIC    SECTIONS    AS    HOMOLOGOUS    FIGURES.  — RELIEF-PERSPECTIVE.     49 


to  that  of  o,   and,   like  the  latter,   S  would    evidently   maintain    its    distance    from    the    vanishing    line 
and   describe   a   circular  arc   about  it.      The   vanishing  line  would   remain   parallel  to   the  axis. 

150.  From    the    foregoing   we   see   that    to    obtain    the  hyperbola,   by   projection   of  a  circle  from  a 
point    in    the    plane    of   the    latter,   we    would    require    simply  a    secant   vanishing  line,  M N  (Fig.   94), 
and   an   axis   of   homology   parallel   to  it.      Take   any   point  P  on  the  vanishing   line    and   join  it  with 
any  point  K  of  the  circle.     PK  meets  the    axis    at   y;    hence    whatever    line  -corresponds    to   PK  must 
also  meet  the  axis   at  y.     OP  is    analogous  to  S  A   of  Fig.  93,  in  that  it  meets   its  corresponding  line 
at  infinity,   i.e.,  is  parallel  to  it.      Therefore  y k,  parallel  to  OP,  corresponds   to   Py,  and  meets  the  ray 
OK  at   k,     corresponding    to   K.      Then  K  joined   with    any    other    point  R  gives  Kz.     Join   z   with  k 
and   prolong   0  R  to  intersect  k  z,   obtaining  r,   another  point   of  the   hyperbola. 

151.  In  Fig.   93,  were  a   tangent   drawn   to    arc  AH B  at  -B,  it  would   meet   the   axis  in  a  point 
which,   like   all   points   on  the  axis,    "corresponds  to    itself."      From    that '  point    the    projection    of   that 
tangent  on  the  lower  plane  would  be  parallel  to  SB,  since  they  are  to  meet  at  infinity.      Or,  if  S  J 
is    parallel   to   the    tangent    at   B,    then  J   will    be    the    projection    of  J1    at    infinity,   where   SJ   meets 
the    tangent;   J  will    be    therefore    one    point    of   the    projection    of   said   tangent    on   the   lower  plane; 
while  another  point  would   be,   as   previously   stated,   that    in   which    the    tangent  at  B  meets  the  axis. 

152.  Analogously   in   Fig.   94,   the  tangents    at  M  and  N  meet  the    axis,   as   at  F  and   E;    but  the 
projectors  OM  and   ON  go   to   points   of   tangency   at   infinity;   M  and   N  are    on    a    "vanishing    line"; 
hence   OM  is   parallel  to   the  tangent  at   infinity,   that    is,  to    the    asymptote    (see    Art.    134)   through  F; 
while  the   other  asymptote   is   a   parallel   through  E  to   ON. 

153.  As    in    Fig.   93   the    projectors   from  S  to    all    points    of  the    arc  above  the  level  of  S  could 
cut  the  lower  plane   only   by  being  produced  to  the  right,  giving  the  right-hand  branch  of  the  hyper- 
bola;  so,  in  Fig.  94,  the  arc  MHN,  above  the  vanishing  line,  gives  the  lower  branch  of  the  hyperbola. 
To   get  a   point    of  the    latter,   as  h,   and    having    already   obtained    any   point    x   of   the    other    branch, 
join  H  with  X  (the   original   of  a>)   and   get  its  intersection,  g,  with   the  axis.      Then  xgh  corresponds 
to  g  X  H,  and  the  ray  0  H  meets  it  at  h,  the  projection  of  H. 

The  cases  should  be  worked  out  in  which  the  vanishing  line  is  tangent  to  the  circle  or  exterior  to  it. 

154.  The    homological    figures    with    which    we    have    been    dealing    were    plane    figures.      But   it    is 
possible  to  have  space  figures   homological  with   each   other. 

In  homological  space  figures  corresponding  lines  meet 
at  the  same  point  in  a  plane,  instead  of  the  same  point 
on  a  line.  A  vanishing  plane  takes  the  place  of  a  van- 
ishing line.  The  figure  that  is  in  homology  with  the 
original  figure  is  called  the  relief -perspective  of  the  latter. 
(See  Art.  11.) 

Remarkably  beautiful  effects  can  be  obtained  by  the 
construction  of  homological  space  figures,  as  a  glance  at 
Fig.  95  will  show.  The  figure  represents  a  triple  row 
of  groined  arches,  and  is  from  a  photograph  of  a  model 
designed  by  Prof.  L.  Burmester. 

Although  not  always  requiring  the  use  of  the  irregular  curve,  and  therefore  not  strictly  the  material 
for  a  topic  in  this  chapter,  its  close  analogy  to  the  foregoing  matter  may  justify  a  few  words  at  this 
point  on  the  construction  of  a  relief-perspective. 

155.  In    Fig.    96   the   plane  P  Q   is    called    the    plane    of   homology    or    picture -plane,    and  —  adopting 
Cremona's   notation — we   will   denote   it   by   ir.      The  vanishing  plane  MN,  or  <£',  is   parallel  to  it.     0  is 


:,  se. 


50 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


the  centre  of  homology  or  perspective- centre.  All  points  in  the  plane  TT  are  their  own  perspectives,  or,  in 
other  words,  correspond  to  themselves.  Therefore  B"  is  one  point  of  the  projection  or  perspective  of 
the  line  A  B,  being  the  intersection  of  A  B  with  TT.  The  line  0  v,  parallel  to  A  B,  would  meet  the 


latter  at  infinity;  therefore  v,  in  the  vanishing  plane  <£',  would  be  the  projection  upon  it  of  the 
point  at  infinity.  Joining  v  with  B",  and  cutting  v  B"  by  rays  OA  and  OB,  gives  A'  B'  as  the  relief- 
perspective  of  A  B.  The  plane  through  0  and  A  B  cuts  it  in  B"  n,  which  is  an  axis  of  homology  for 
AB  and  A' B',  exactly  as  mn  in  Fig.  92  is  for  AlBl  and  A^B,. 


;>x7^$j::W:;f--,i...  'I 

i,  \>,  m\-f- 


As    D  C  in    Fig.    96    is    parallel    to    A  B,    a    parallel    to    it    through    0    is    again    the    line    0  v. 


LINK-MOTION    CURVES.  51 

The  trace  of  D  C  on  v  is  C ".  Joining  v  with  C"  and  cutting  v  C"  by  rays  0  D,  0  C,  obtains 
D'  C'  in  the  same  manner  as  A'  B'  was  derived.  The  originals  of  A'  B'  and  C'  D'  are  parallel  lines; 
but  we  see  that  their  relief -perspectives  meet  at  v.  The  vanishing  plane  is  therefore  the  locus*  of 
the  vanishing  points  of  lines  that  are  parallel  on  the  original  object,  while  the  plane  of  homology 
is  the  locus  of  the  axes  of  homology  of  corresponding  lines;  or,  differently  stated,  any  line  and  its 
relief -perspective  will,  if  produced,  meet  on  the  plane  of  homology. 

156.  Fig.   97   is  inserted  here  for  the  sake  of  completeness,  although  its  study  may  be  reserved,  if 
necessary,   until    the    chapter    on    projections    has    been    read.      In    it   a    solid    object    is    represented    at 
the  left,   in   the   usual   views,   plan   and   elevation;    G  L  being  the  ground  line  or  axis   of  intersection   of 
the   planes   on   which   the   views   are   made.      The   planes   ?r  and   <f>    are   interchanged,  as  compared  with 
their   positions   in   Fig.    96,   and  they   are   seen   as   lines,   being    assumed    as    perpendicular  to   the   paper. 
The  relief-  perspective  appears  between  them,   in  plan  and   elevation. 

The  lettering  of  A  B  and  D  C,  and  the  lines  employed  in  getting  their  relief- perspectives,  being 
identical  with  the  same  constructions  in  Fig.  96,  ought  to  make  the  matter  clear  at  a  glance  to  all 
who  have  mastered  what  has  preceded. 

Burmester's  Grundzuge  der  Relief- Perspective  and  Wiener's  Darstellende  Geometric  are  valuable  reference 
works  on  this  topic  for  those  wishing  to  pursue  its  study  further;  but  for  special  work  in  the 
line  of  homological  plane  figures  the  student  is  recommended  to  read  Cremona's  Projective  Geometry 
and  Graham's  Geometry  of  Position,  the  latter  of  which  is  especially  valuable  to  the  engineer  or  architect, 
since  it  illustrates  more  fully  the  practical  application  of  central  projection  to  Graphical  Statics. ^-^==^~ 

LINK-MOTION     CURVES 

157.  Kinematics  is   the    science    which    treats    of  pure  motion,  regardless  of  the  cause  or  the  results  of 
the  motion. 

It  is  a  purely  kinematic  problem  if  we  lay  out  on  the  drawing-board  the  path  of  a  point  on 
the  connecting-rod  of  a  locomotive,  or  of  a  point  on  the  piston  of  an  oscillating  cylinder,  or  of  any 
point  on  one  of  the  moving  pieces  of  a  mechanism.  Such  problems  often  arise  in  machine  design, 
especially  in  the  invention  or  modification  of  valve  -  motions. 

Some  of  the  motion -curves  or  point -paths  that  are  discovered  by  a  study  of  relative  motion  are 
without  special  name.  Others,  whose  mathematical  properties  had  already  been  investigated  and  the 
curves  dignified  with  names,  it  was  later  found  could  be  mechanically  traced.  Among  these  the 
most  familiar  examples  are  the  Ellipse  and  the  Lemniscate,  the  latter  of  which  is  employed  here  to 
illustrate  the  general  problem. 

The  moving  pieces  in  a  mechanism  are  rigid  and  inextensible,  and  are  always  under  certain 
conditions  of  restraint.  "Conditions  of  restraint"  may  be  illustrated  by  the  familiar  case  of  the  con- 
necting-rod of  the  locomotive,  one  end  of  Avhich  is  always  attached  to  the  driving-wheel  at  the 
crank -pin  and  is  therefore  constrained  to  describe  a  circle  about  the  axle  of  that  wheel,  while  the 
other  end  of  the  rod  must  move  , in  a  straight  line,  being  fastened  by  the  "wrist-pin"  to  the  "cross- 
head,"  which  slides  between  straight  "guides."  The  first  step  in  tracing  .a  point -path  of  any 
mechanism  is  therefore  the  determination  of  the  fixed  points,  and  a  general  analysis  of  the  motion. 


"Locus  is  the  Latin  for  place;  and  in  rather  untechnical  language,  although  in  the  exact  sense  in  which  it  is  used  mathe- 
matically, we  may  say  that  the  loms  of  points  or  lines  is  the  place  where  you  may  expect  to  find  them  under  their  conditions 
of  restriction.  For  example,  the  surface  of  a  sphere  is  the  locus  of  all  points  equidistant  from  a  fixed  point  (Its  centre).  The 
locus  of  a  point  moving  in  a  plane  so  as  to  remain  at  a  constant  distance  from  a  given  fixed  point,  is  a  circle  having  the 
latter  point  as  its  centre. 


52 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


158.  We  have  given,  in  Fig.  98,  two  links  or  bars,  MN  and  S  P,  fastened  at  N  and  P  by 
pivots  to  a  third  link,  NP,  while  their  other  extremities  are  pivoted  on  stationary  axes  at  M  and 
S.  The  only  movement  possible  to  the  point  N  is  therefore  in  a  circle  about  M;  while  P  is 
equally  limited  to  circular  motion  about  S.  The  points  on  the  link  NP,  with  the  exception  of  its 


2  MN   2  MS 


THE   LEMNISCATE 

AS  A 
LINK-MOTION   CURVE 


extremities,  have  a  compound  motion,  in  curves  whose  form  it  is  not  easy  to  predict  and  which 
differ  most  curiously  from  each  other.  The  figure-of-eight  curve  shown,  otherwise  the  "Lemniscate 
of  Bernoulli,"  is  the  point -path  of  Z,  the  link  NP  being  supposed  prolonged  by  an  amount,  P  Z, 
equal  to  NP.  Since  NP  is  constant  in  length,  if  N  were  moved  along  to  F,  the  point  P  would 
have  to  be  at  a  distance  NP  from  F, and  also  on  the  circle  to  which  it  is  confined;  therefore  its 
new  position  /,  is  at  the  intersection  of  the  circle  Psr  by  an  arc  of  radius  PN,  centre  F.  Then 
Ff,  prolonged  by  an  amount  equal  to  itself,  gives  /, ,  another  point  of  the  Lemniscate,  and  to  which 
Z  has  then  moved.  All  other  positions  are  similarly  found. 

If  the  motion  of  N  is  toward  D  it  will  soon  reach  a  limit,  A,  to  its  further  movement  in  that 
direction,  arriving  ther.e  at  the  instant  that  P  reaches  a,  when  NP  and  PS  will  be  in  one  straight 
line,  SA.  In  this  position  any  movement  of  P  either  side  of  a  will  drag  N  back  over  its  former 
path;  and  unless  P  moves  to  the  left,  past  a,  it  would  also  retrace  its  path.  P  reaches  a  similar 
"dead  point"  at  v. 

To  obtain  a  Lemniscate  the  links  NP  and  PS  had  to  be  equal,  as  also  the  distance  MS 
to  MN.  Bv  varying  the  proportions  of  the  links,  the  point- paths  would  be  correspondingly  affected. 


INSTANTANEOUS    CENTRES.  — CENT  RO 1 DS. 


53 


By  tracing  the  path  of  a  point  on  PN  produced,  and  as  far  from  N  as  Z  is  from  P,  the 
student  will  obtain  an  interesting  contrast  to  the  Lemniscate. 

If  M  and  S  were  joined  by  a  link,  and  the  latter  held  rigidly  in  position,  it  would  have  been 
called  the  fixed  link;  and  although  its  use  would  not  have  altered  the  motions  illustrated,  and  it  is 
not  essential  that .  it  should  be  drawn,  yet  in  considering  a  mechanism  as  a  whole,  the  line  joining 
the  fixed  centres  always  exists,  in  the  imagination,  as  a  link  of  the  complete  system. 

INSTANTANEOUS     CENTRES.  —  CENTROIDS. 

159.  Let  us  imagine  a  boy  about  to  hurl  a  stone  from  a  sling.  Just  before  he  releases  it 
he  runs  forward  a  few  steps,  as  if  to  add  a  little  extra  impetus  to  the  stone.  While  taking  those  few 
steps  a  peculiar  shadow  is  cast  on  the  road  by  the  end  of  the  sling,  if  the  day  is  bright.  The 


boy  moves  with  respect  to  the  earth;  his  hand  moves  in  relation  to  himself,  and  the  end  of  the 
sling  describes  a  circle  about  his  hand.  The  last  is  the  only  definite  element  of  the  three,  yet  it 
is  sufficient  to  simplify  otherwise  difficult  constructions  relating  to  the  complex  curve  which  is 
described  relatively  to  the  earth. 


54  THEORETICAL    AND    PRACTICAL    GRAPHICS. 

A  tangent  and  a  normal  to  a  circle  are  easily  obtained,  the  former  being,  as  need  hardly  be 
stated  at  this  point,  perpendicular  to  the  radius  at  the  point  of  tangency,  while  the  normal  simply 
coincides  in  direction  with  such  radius.  If  the  stone  were  released  at  any  instant  it  would  fly  off 
in  a  straight  line,  tangent  to  the  circle  it  was  describing  about  the  hand  as  a  centre;  but  such  line 
would,  at  the  instant  of  release,  be  tangent  also  to  the  compound  curve.  If,  then,  we  wish  a  tangent 
at  a  given  point  of  any  •  curve  generated  by  a  point  in  motion,  we  have  but  to  reduce  that  motion 
to  circular  motion  about  some  moving  centre;  then,  joining  the  point  of  desired  tangency  with  the — 
at  that  instant — position  of  the  moving  centre,  we  have  the  normal,  a  perpendicular  to  which  gives 
the  tangent  desired. 

A   centre   which   is   thus   used  for  an  instant  only  is   called   an  instantaneous  centre. 

160.  In   Fig.    99   a  series   of  instantaneous   centres  are   shown    and    an    important   as   well  as   inter- 
esting fact  illustrated,  viz.,  that   every  moving  piece   in  a   mechanism    might  be  rigidly  attached  to  a 
certain    curve,   and    by  the    rolling   of  the   latter   upon   another    curve    the    link    might  be  brought   into 
all  the   positions   which   its   visible   modes   of  restraint   compel   it   to   take. 

161.  In  the   "Fundamental"   part   of  Fig.   99  A  B  is   assumed   to   be   one   position   of  a  link.      We 
next    find    it,    let    us    suppose,    at  A' B',  A  having    moved    over  A  A',    and    B    over   BB'.     Bisecting 
A  A'    and   B  B'    by    perpendiculars    intersecting    at    0,    and    drawing    0  A,     OA',     OB    and     OB',    we 
have  A  0  A'  =  6t  =  B  OB',   and    0  evidently    a    point    about    which,    as    a    centre,    the    turning    of  AB 
through   the   angle   0,   would    have    brought    it   to  A' B'.      Similarly,   if  the   next    position  in   which   we 
find  AB  is  A"  B",    we    may   find    a    point    s    as    the    centre    about    which    it    might    have    turned    to 
bring   it  there;    the   angle  being  02,   probably   different   from  0,.      N  and  m  are   analogous   to   0  and   s. 

If  Os'  be  drawn  equal  to  Os  and  making  with  the  latter  an  angle  0,,  equal  to  the  angle 
A  OA',  and  if  Os  were  rigidly  attached  to  A  B,  the  latter  would  be  brought  over  to  A' B'  by 
bringing  0  s'  into  coincidence  with  0  s.  In  the  same  manner,  if  we  bring  s'  n'  upon  s  n  through 
an  angle  02  about  s,  then  the  next  position,  A"  B",  would  be  reached  by  A  B.  0' s' n' m'  is  then 
part  of  a  polygon  whose  rolling  upon  Osnm  would  bring  A  B  into  all  the  positions  shown,  provided 
the  polygon  and  the  line  were  so  attached  as  to  move  as  one  piece.  Polygons  whose  vertices  are 
thus  obtained  are  called  central  polygons. 

If  consecutive  centres  were  joined  we  would  have  curves,  called  centroids*,  instead  of  polygons; 
the  one  corresponding  to  Osnm  being  called  the  fixed,  the  other  the  rolling  centroid.  The  perpen- 
dicular from  0  upon  A  A '  is  a  normal  to  that  path.  But  were  A  to  move  in  a  circle,  the  normal 
to  its  path  at  any  instant  would  be  simply  the  radius  to  the  position  of  A  at  that  instant. 

If,  then,  both  A  and  B  were  moving  in  circular  paths,  we  would  find  the  instantaneous  centre 
at  the  intersection  of  the  normals  (radii)  at  the  points  A  and  B. 

162.  In    Fig.    98    the    instantaneous    centre    about   which   the  whole  link  NP  is   turning,  is   at  the 
intersection   of  radii  M  N  and  SP  (produced);    and   calling    it  X  we  would    have  XZ   for    the    normal 
at   Z  to   the    Lemniscate. 

163.  The   shaded   portions   of  Fig.   99   illustrate   some   of  the   forms   of  centroids. 

The  mechanism  is  of  four  links,  opposite  links  equal.  Unlike  the  usual  quadrilateral  fulfilling 
this  condition,  the  long  sides  cross,  hence  the  name  "anti- parallelogram." 

The  "fixed  link  (a)"  corresponds  to  MS  of  Fig.  98,  and  its  extremities  are  the  centres  of 
rotation  of  the  short  links,  whose  ends,  /  and  /,,  describe  the  dotted  circles. 

For    the    given    position    T  is    evidently   the    instantaneous   centre.      Were   a   bar  pivoted   at   T  and 


*Reuleaux'  nomenclature;   also  called  centrodes  by  a  number  of  writers  on  Kinematics. 


TROCHOIDS.  55 

fastened    at    right    angles    to    "moving    link    (a),"    an    infinitesimal   turning   about   T  would  move  "link 
(a)"  exactly  as   under  the  old  conditions. 

By  taking  "link  (a)"  in  all  possible  positions,  and,  for  each,  prolonging  the  radii  through  its 
extremities,  the  points  of  the  fixed  centroid  are  determined.  Inverting  the  combination  so  that 
"moving  link  (a)"  and  its  opposite  are  interchanged,  and  proceeding  as  before,  gives  the  points  of 
"rolling  centroid  (a)." 

These  centroids  are  branches  of  hyperbolas  having  the  extremities   of  the  long  links  as  foci. 
By    holding   a    short    link    stationary,    as     "fixed    link    (b),"    an    elliptical    fixed    centroid    results; 
"rolling  centroid  (b)"  being  obtained,  as  before,  by  inversion.      The  foci  are  again  the  extremities  of 
the  fixed  and  moving  links. 

Obviously,  the  curved  pieces  represented  as  screwed  to  the  links  would  not  be  employed  in  a 
practical  construction,  and  they  are  only  introduced  to  give  a  more  realistic  effect  to  the  figure  and 
possibly  thereby  conduce  to  a  clearer  understanding  of  the  subject. 

164.  It    is    interesting    to    notice    that    the    Lemniscate    occurs    here    under    new    conditions,    being 
traced   by   the  middle  point   of  "moving  link   (a)." 

The  study  of  kinematics  is  both  fascinating  and  profitable,  and  it  is  hoped  that  this  brief  glance 
at  the  subject  may  create  a  desire  on  the  part  of  the  student  to  pursue  it  further  in  such  works  as 
Reauleaux'  Kinematics  of  Machinery  and  Burm ester's  Lehrbuch  der  Kinematik. 

165.  Before  leaving   this   topic   the  important  fact  should   be  stated,  which   now  needs  no  argument 
to    establish,    that    the    instantaneous    centre,    for    any    position    of    a    moving     piece,    is     the    point    of 
contact  of  the   rolling   and   fixed   centroids.      We   shall    have    occasion   to   use   this    principle   in   drawing 
tangents   and   normals   to   the 

TROCHOIDS 

which   are   the   principal   Roulettes,   or  roll -traced  curves,   and   which   may   be   defined   as   follows:  — 

If,  in  the  same  plane,  one  of  two  circles  roll  upon  the  other  without  sliding,  the  path  of  any 
point  on  a  radius  of  the  rolling  circle  or  on  the  radius  produced  is  a  trochoid. 

166.  The   Cycloid.      Since    a   straight   line    may   be    considered    a   circle    of  infinite  radius,  the  above 
definition   would   include   the   curve  traced   by   a  point   on  the   circumference   of   a  locomotive   wheel   as 
it  rolls   along   the   rail,   or  of  a   carriage   wheel   on    the   road.      This    curve   is    known   as   a   cycloid*  and 
is  shown  in    T  n  a  b  c,  Fig.   100.      It  is  the  proper  outline  for   a   portion    of   each    tooth    in    a    certain 
case  of  gearing,  viz.,  where  one  wheel  has  an  infinite  radius,  that  is,  becomes  a  "rack." 

Were  T6  a  ceiling -corner  of  a  room,  and  Ta  the  diagonally  opposite  floor-corner,  a  weight  would 
slide  from  T&  to  Tu  more  quickly  on  guides  curved  in  cycloidal  shape  than  if  shaped  to  any  other 
curve,  or  if  straight.  If  started  at  s,  or  any  other  point  of  the  curve,  it  would  reach  Ta  as  soon 
as  if  started  at  Te. 

167.  In    beginning    the    construction    of   the    cycloid    we    notice,  first,  that  as    T  V D   rolls    on  the 
straight  line  A  B,  the  arrow  DRT  will  be  reversed   in  position    (as  at  -D5  T6)    as    soon    as    the   semi- 
circumference    T3Z)    has    had    rolling    contact    with  A  B.      The    tracing   point    will    then    be    at    T6,    its 
maximum   distance   from  A  B. 

When  the  wheel  has  rolled  itself  out  once  upon  the  rail,  the  point  T  will  again  come  in  contact 
with  the  rail,  as  at  jT12. 

*"  Although  the  invention  of  the  cycloid  is  attributed  to  Galileo,  it  is  certain  that  the  family  of  curves  to  which  it  belongs 
had  been  known  and  some  of  the  properties  of  such  curves  investigated,  nearly  two  thousand  years  before  Galileo's  time,  if 
not  earlier.  For  ancient  astronomers  explained  the  motion  of  the  planets  by  supposing  that  each  planet  travels  uniformly 
round  a  circle  whose  centre  travels  uniformly  around  another  circle."— Proctor,  Geometry  of  Cycloids. 


56 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


The    distance    TT12    evidently    equals    2*r,    when    r=TR.      We    also    have 

If  the  semi-circumference  T3D  (equal  to  irr)  be  divided  into  any  number  of  equal  parts,  and 
also  the  path  of  centres  RRe  (again  =  irr)  into  the  same  number  of  equal  parts,  then  as  the  points 
1,  2,  etc.,  come  in  contact  with  the  rail,  the  centre  R  will  take  the  positions  J2j,JS2,  etc.,  directly 
above  the  corresponding  points  of  contact.  A  sufficient  rolling  of  the  wheel  to  br-ing  point  2  upon 
A  B  would  evidently  raise  T  from  its  original  position  to  the  former  level  of  2.  But  as  T  must 
always  be  at  a  radius'  distance  from  R,  and  the  latter  would  by  that  time  be  at  Rlt  we  would  find 
T  located  at  the  intersection  («)  of  the  dotted  line  of  level  through  2  by  an  arc  of  radius  R  T, 
centre  R  2.  Similarly  for  other  points. 

The  construction,  summarized,  involves  the  drawing  of  lines  of  level  through  equidistant  points  of 
division  on  a  semi-circumference  of  the  rolling  circle,  and  their  intersection  by  arcs  of  constant  radius 
(that  of  the  rolling  circle)  from  centres  which  are  the  successive  positions  taken  by  the  centre  of  the 
rolling  circle. 

It  is  worth  while  calling  attention  to  a  point  occasionally  overlooked  by  the  novice,  although 
almost  self-evident,  that,  in  the  position  illustrated  in  the  figure,  the  point  T  drags  behind  the  centre 
R  until  the  latter  reaches  Re,  when  it  passes  and  goes  ahead  of  it.  From  R1  the  line  of  level 
through  5  could  be  cut  not  alone  at  c  by  an  arc  of  radius  cR,  but  also  in  a  second  point; 
evidently  but  one  of  these  points  belongs  to  the  cycloid,  and  the  choice  depends  upon  the  direction 
of  turning,  and  upon  the  relative  position  of  the  rolling  centre  and  the  moving  point.  This  matter 
requires  more  thought  in  drawing  trochoidal  curves  in  which  both  circles  have  finite  radii,  as  will 
appear  later. 


-.  ±OO. 


168.  Were   points  T6  and   T12  given,  and  the  semi -cycloid   T6  T12  desired,  we  can   readily  ascertain 
the   "base,"   A  B,  and   generating   circle,   as    follows:     Join    T6  with    T12;    at  any   point   of  such  line,   as 
x,   erect  a   perpendicular,  xy;    from   the   similar  triangles  xyTtl  and    T^D^T^,  having   angle   <f>  common 
and   angles   0  equal,   we   see  that 

xy:xTM::  T6.D5 :  Z>5  T12 : :  2r  :  TTT- : :  2  :  *•: :  1 :  £;    or,   very   nearly,   as   14:22. 

a 

If,  then,  we  lay  off  x  T]2  equal  to  twenty -two  equal  parts  on  any  scale,  and  a  perpendicular,  xy, 
fourteen  parts  of  the  same  scale,  the  line  y  Tn  will  be  the  base  of  the  desired  curve;  while  the 
diameter  of  the  generating  circle  will  be  the  perpendicular  from  Te  to  y  Tu  prolonged. 

169.  To   draw  the  tangent  to   a    cycloid    at    any  point  is   a   simple  matter,   if  we  see  the  analogy 
between  the  point  of  contact  of  the   wheel   and    rail    at   any   instant,   and  the   hand  used   in  the  former 
illustration   (Art.    159).      At   any   one  moment   each   point   on    the    entire    wheel    may   be    considered    as 
describing  an  infinitesimal   arc    of   a   circle   whose   radius   is   the  line  joining  the   point   with   the   point 
of   contact    on    the    rail.      The    tangent    at   N,    for    example,    (Fig.    100),    would   be    t  N,    perpendicular 
to  the  normal,   No,  joining   N  with   o;    the    latter    point    being    found    by   using   N  as    a    centre    and 


THE    CYCLOID.— COMPANION    TO     THE    CYCLOID. 


57 


cutting  AS  by  an  arc  of  radius  equal  to  m  I,  in  which  m  is  a  point  at  the  level  of  N  on  any 
position  of  the  rolling  circle,  while  I  is  the  corresponding  point  of  contact.  The  point  o  might  also 
have  been  located  by  the  following  method:  Cut  the  line  of  centres  by  an  arc,  centre  N,  radius 
TR;  o  would  obviously  be  .vertically  below  the  position  of  the  rolling  centre  thus  determined. 

170.  The  Companion  to  the   Cycloid.     The  kinematic   method   of  drawing  tangents,  just  applied,  was 
devised   by   Roberval,   as   also   the   curve    named   by   him   the    "Companion    to'  the    Cycloid,"    to    which 
allusion   has   already   been  made   (Art.    120)   and   which   was  invented   by  him   in   1634  for  the  purpose 
of   solving    a    problem    upon    which    he    had    spent    six    years    without    success,   and    which   had   foiled 
Galileo,  viz.,  the  calculating  of  the  area  between  a  cycloid  and  its  base.     Galileo  was  reduced  to  the 
expedient   of    comparing  the   area   of   the   cycloid    with    that    of    the    rolling    circle    by   weighing    paper 
models    of   the    two    figures.      He    concluded    that    the    area   in    question   was   nearly  but  not    exactly 
three  times  that   of  the   rolling  circle.      That  the  latter  would   have  been  the  correct  solution  may  be 
readily   shown   by  means   of  the   "Companion,"   as   will   be   found   demonstrated  in   Art.    172. 

171.  Suppose  two  points  coincident  at  T  (Fig.  101)  and  starting  simultaneously  to  generate  curves, 
the  first  of  these  points  to  trace  the  cycloid  during  the  rolling  of  circle   TVD,  while  the  second  is  to 
move  independently  of  the  circle  and  so  as  to  be  always  at  the  level  of  the  point  tracing  the  cycloid, 
yet  at  the  same  time   vertically  above  the   point   of  contact   of  the   circle   and  base.      This   makes   the 
second    point    always    as    far   from    the   initial    vertical    diameter,   or  axis,   of  the   cycloid,   as   the  length 
of  the   arc   from   T  to   whatever  level   the  tracing   point    of  the  latter   has    then    reached;    that  is,   MA 
equals   arc    THs;    RO  equals   quadrant   Tsy. 

Adopting  the  method  of  Analytical  Geometry,  and  using  0  as  the  origin,  we  may  reach  any 
point,  A,  on  the  curve,  by  co-ordinates,  as  Ox,  x A,  of  which  the  horizontal  is  called  an  abscissa,  the 
vertical  an  ordinate.  By  the  preceding  construction  Ox  equals  arc  sfy,  while  xA  equals  siv — the 
sine  of  the  same  arc.  The  "Companion"  is  therefore  a  curve  of  sines  or  sinusoid,  since,  starting  from 
0,  the  abscissas  are  equal  to  or  proportional  to  the  arc  of  a  circle,  while  the  ordinates  are  the  sines 
of  those  arcs.  It  is  also  the  orthographic  projection  of  a  45° -helix. 

This  curve  is  particularly  interesting  as  "expressing  the  law  of  the  vibration  of  perfectly  elastic 
solids;  of  the  vibratory  movement  of  a  particle  acted  upon  by  a  force  which  varies  directly  as  the 
distance  from  the  origin;  approximately,  the  vibratory  movement  of  a  pendulum;  and  exactly  the, 
law  of  vibration  of  the  so-called  mathematical  pendulum."*  (See  also  Art.  356). 

172.  From    the     symmetry    of   the 
sinusoid    with    respect    to  RR6  and    to 
0,    we    have     area     TAOR  =  ECORe; 
adding    area   D  EL  0  R    to    both    mem- 
bers   we    have    the    area    between    the 
sinusoid    and    TD    and    DE    equal    to 
the  rectangle  R  E,  or  one-half  the  rect- 
angle    D  E  K  T;     or    to    ^  *  r  X  2  r  = 

Trr2,  the  area   of  the  rolling   circle. 

As  TA  C  E  is  but  half  of  the  entire  sinusoid,  it  is  evident  that  the  total  area  below  the  curve 
is  twice  that  of  the  generating  circle. 

The  area  between  the  cycloid  and  its  "companion"  remains  to  be  determined,  but  is  readily 
ascertained  by  noting  that  as  any  point  of  the  latter,  as  A,  is  on  the  vertical  diameter  of  the  circle 

*  Wood,  Elements  of  Co-ordinate  Geometry,  p.  209. 


58 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


passing  through  the  then  position  of  the  tracing  point,  as  a,  the  distance,  A  a,  between  the  two 
curves  at  any  level,  is  merely  the  semi-chord  of  the  rolling  circle  at  that  level.  But  this,  evidently, 
equals  Ms,  the  semi-chord  at  the  same  level  on  the  equal  circle.  The  equality  of  Ms  and  A  a 
makes  the  elementary  rectangles  Mss1ml  and  AAlala  equal;  and  considering  all  the  possible 
similarly  -  constructed  rectangles  of  infinitesimal  altitude,  the  sum  of  those  on  semi-chords  of  the 
rolling  circle  would  equal  the  area  of  the  semi-circle  TDy,  which  is  therefore  the  extent  of  the  area 
between  the  two  curves  under  consideration. 

The  figure  showing  but  half  of  a  cycloid,  the  total  area  between  it  and  its  "companion"  must 
be  that  of  the  rolling  circle.  Adding  this  to  the  area  between  the  "companion"  and  the  base 
makes  the  total  area  between  cycloid  and  base  equal  to  three  times  that  of  the  rolling  circle. 

173.  The  paths  of  points  carried  by  and  in  the  plane  of  the  rolling  circle,  though  not  on  its 
circumference,  are  obtained  in  a  manner  closely  analogous  to  that  employed  for  the  cycloid. 

In  Fig.  102  the  looped  curve,  traced  by  the  arrow-point  while  the  circle  CHM  rolls  on  the 
base  A  B,  is  called  the  Curtate  Trochoid.  To  obtain  the  various  positions  of  the  tracing  point  T 
describe  a  circle  through  it  from  centre  R.  On  this  circle  lay  off  any  even  number  of  equal  arcs,  and 
draw  radii  from  R  to  the  points  of  division;  also  "lines  of  level"  through  the  latter.  The  radii 
drawn  intercept  equal  arcs  on  the  rolling  circle  CHM,  whose  straight  equivalents  are  next  laid  off  on 
the  path  of  centres,  giving  Rlt  R2,  etc.  While  the  first  of  these  arcs  rolls  upon  A  B,  the  point  T  turns 
through  the  angle  TR  1  about  R,  and  reaches  the  line  of  level  through  point  1.  But  T  is  always  at  the 
distance  R  T  (called  the  tracing  radius)  from  R;  and,  as  R  has  reached  R}  in  the  rolling  supposed,  we 
will  find  T1  —  the  new  position  of  T  —  by  an  arc  from  Rlt  radius  TR,  cutting  said  line  of  level. 


-.  1O3. 


___4 1— -(-—4 ,— -, 


After    what   has    preceded,  the   figure   may  be  assumed   to   be  self-interpreting,  each  position  of  T 
having  been  joined  with  the  position  of  R  which   determined  it. 

174.  Were  a  tangent  wanted  at  any  point,  as    T,,  we  have,  as  before,  to  determine  the  point  of 
contact  of  rolling  circle  and  line    when    T  reached    T,,  and    use    it    as    an    instantaneous    centre.      7", 
was   obtained   from   .R,;    and   the   point  of  contact  must    have   been   vertically   below  the  latter  and  on 
A  B.      Joining  such  point  to    T,  gives  the  normal,  from  which  the  tangent  follows  in  the  usual  way. 

175.  The  Prolate   Trochoid.     Had  we  taken  a    point  inside  of  the   circle  CHM  and   constructed  its 
path,  the  only  difference  between  it  and  the  curve  illustrated  would  have  been  in  the  name  and  the 


HYPO-,    EPI-    AND    PERI-TROCHOIDS. 


59 


shape  of  the  curve.  An  undulating,  wavy  path  would  have  resulted,  called  the  prolate  trochoid;  but, 
as  before,  we  would  have  described  a  circle  through  the  tracing  point;  divided  it  into  equal  parts; 
drawn  lines  of  level,  and  cut  them  by  arcs  of  constant  radius,  using  as  centres  the  successive 
positions  of  R.  A  bicycle  pedal  describes  a  prolate  trochoid  relatively  to  the  earth. 

HYPO-,      KI'I-      AND      PERI-TROCHOIDS.  , 

176.  Circles  of  finite  radius  can  evidently  be  tangent  in  but  two  ways  —  either  externally,  or 
Internally;  if  the  latter,  the  larger  may  roll  on  the  one  within  it,  or  the  smaller  may  roll  inside 
the  larger.  When  a  small  circle  rolls  within  a  larger,  the  radius  of  the  latter  may  be  greater  than 
the  diameter  of  the  rolling  circle,  or  may  equal  it,  or  be  smaller.  On  account  of  an  interesting 
property  of  the  curves  traced  by  points  in  the  planes  of  such  rolling  circles,  viz.,  their  capability  of 
being  generated,  trochoidally,  in  two  ways,  a  nomenclature  was  necessary  which  would  indicate  how 
each  curve  was  obtained.  This  is  included  in  the  tabular  arrangement  of  names  below,  and  which 
was  the  outcome  of  an  investigation*  made  by  the  writer  in  1887  and  presented  before  the  American 
Association  for  the  Advancement  of  Science.  In  accepting  the  new  terms,  advanced  at  that  time, 
Prof.  Francis  Reuleaux  suggested  the  names  Ortho- cycloids  and  Cydo-orthoids  for  the  classes  of  curves 
of  which  the  cycloid  and  involute  are  respectively  representative;  orthoids  being  the  paths  of  points 
in  a  fixed  position  with  respect  to  a  straight  line  rolling  upon  any  curve,  and  cydo-orthoid  therefore 
implying  a  circular  director  or  base- curve.  These  'appropriate  terms  have  been  incorporated  in  the 
table. 

For  the  last  column  a  point  is  considered  as  within  the  rolling  circle  of  infinite  radius  when  on 
the  normal  to  its  initial  position,  and  on  the  side  toward  the  centre  of  the  fixed  circle. 

As  will  be  seen  by  reference  to  the  Appendix,  the  curves  whose  names  are  preceded  by  the 
same  letter  may  be  identical.  Hence  the  terms  curtate  and  prolate,  while  indicating  whether  the 
tracing  point  is  beyond  or  within  the  circumference  of  the  rolling  circle,  give  no  hint  as  to  the 
actual  form  of  the  curves. 

In   the  table,   R  represents   the   radius   of  the   rolling   circle,   F  that   of  the  fixed  circle. 

NOMENCLATURE     OF     TROCHOIDS. 


Position  of 
Tracing 
or 
Describing 
Point. 

Circle  rolling 
upon 
Straight  Line. 
F  =  oo 

Circle  rolling  upon  circle. 

Straight    Line 
rolling    upon 
Circle.     R  =  oo 

External 

contact. 

Internal  contact. 

Larger  Circle 
rolling. 

Smaller  circle  rolling 

Ortho-cycloids. 

Epitrochoids. 

->  R  >  F. 

2  R  <  F. 

2  R  =  F. 

Cyclo-orthoids. 

Peritrochoids. 

Major 
Hypotrochoids. 

Minor 
Hypotrochoids. 

Medial 
Hypotrochoids. 

On    circumference 
of  rolling  circle. 

Cycloid. 

(a)    Epicycloid. 

(a)  Pericycloid. 

(d)        Major 
Hypocycloid. 

(d)         Minor 
Hypocycloid. 

Straight 
Hypocycloid. 

Involute. 

Within 
Circumference. 

Prolate 

Trochoid. 

(b)      Prolate 
Epitrochoid. 

(c)      Prolate              (e)   Major  Prolate 
Peritrochoid.             Hypotrochoid. 

(f)    Minor  prolate 
Hypotrochoid. 

(g)       Prolate 
Elliptical 
Hypotrochoid. 

Prolate 
Cyclo-orthoid. 

Without 
Circumference. 

(f)         Major 
Curtate              (c)      Curtate             (b)     Curtate                         Curtate 
Trochoid.                 Epitrochoid.              Peritrochoid               Hypotrochoid. 

(e)         Minor 
Curtate 
Hypotrochoid. 

(g)       Curtate 
Elliptical                       Curtate 
Hypotrochoid.            Cvclo-orthoid. 

177.  From  the  above  we  see  that  the  prefix  epi  (over  or  upon)  denotes  the  curves  resulting 
from  external  contact;  hypo  (under')  those  of  internal  contact  with  smaller  circle  rolling;  while  peri 
(about)  indicates  the  third  possibility  as  to  rolling. 


*  Re-printed   in  substance   in    the   Appendix. 


60 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


r-  1O3. 


178.     The    construction    of   these    curves    is    in    closest    analogy    to    that    of   the    cycloid.      If,    for 
example,   we    desire    a    major    hypocycloid,  we    first    draw    two    circles,   mVP,  mxL,   (Fig.  103),  tangent 

internally,  of  which  the  rolling  circle  has  its  di- 
ameter greater  than  the  radius  of  the  fixed  circle. 
Then,  as  for  the  cycloid,  if  the  tracing -point  is  P, 
we  divide  the  semi -circumference  m  V  P  into  equal 
parts,  and  from  the  fixed  centre,  F,  describe  circles 
through  the  points  of  division,  as  those  through 
1,  2,  3,  4  and  5.  These  replace  the  "lines  of  level" 
of  the  cycloid,  and  may  be  called  circles  of  distance, 
as  they  show  the  varying  distances  of  the  point  P 
from  F,  for  definite  amounts  of  angular  rotation  of 
the  former.  For  if  the  circle  PVm  were  simply 
to  rotate  about  R,  the  point  P  would  reach  m 
during  a  semi -rotation,  and  would  then  be  at  its 
maximum  distance  from  F.  After  turning  through 
the  equal  arcs  P-l,  1-2,  etc.,  its  distances  from 
F  would  be  Fa  and  Fb  respectively.  If,  however, 
the  turning  of  P  about  R  is  due  to  the  rolling  of 
circle  PVm  upon  the  arc  mxz,  then  the  actual 
position  of  P,  for  any  amount  of  turning  about  R,  is  determined  by  noting  the  new  position  of  R, 
due  to  such  rolling,  as  B1}  R^,  etc.,  and  from  it  as  a  centre  cutting  the  proper  circle  of  distance 
by  an  arc  of  radius  R  P. 

Since  the  radius  of  the  smaller  circle  is  in  this  case  three -fourths  that  of  the  larger,  the  angle 
mFz  (135°),  at  the  centre  of  the  latter,  intercepts  an  arc,  mxz,  equal  to  the  180° -arc,  m  V P,  on  the 
smaller  circle;  for  equal  arcs  on  unequal  circles  are  subtended  by  angles  at  the  centre  which  are  inversely 
proportional  to  the  radii.  As  a  proportion  we  would  have  Fm:Rm::  180°  :  135°.  (In  an  inverse 
proportion  between  angles  and  radii,  in  two  circles,  the  "means"  must  belong  to  one  circle  and  the 
"  extremes  "  to  the  other). 

While  arc  m  V P  rolls  upon  arc  mxz,  the  centre  R  will  evidently  move  over  circular  arc  R — R6. 
Divide  mxz  into  as  many  equal  parts  as  mV P  and  draw  radii  from  F  to  the  points  of  division ; 
these  cut  the  path  of  centres  at  .the  successive  positions  of  R.  When  arc  m  5-4,  for  example,  has 
rolled  upon  its  equal  muv,  then  R  will  have  reached  R2;  P  will  have  turned  about  R  through 
angle  PR2  =  mR4,  and  will  be  at  n,  the  intersection  of  b/g — the  circle  of  distance  through  2 — by 
an  arc,  centre  R2,  radius  R  P.  Similarly  for  other  points. 

179.  General  solution  for  all  trochoidal  curves,  illustrated  by  epi-  and  peri-trochoids.  To  trace  the 
path  of  any  point  on  the  circumference  of  a  circle  so  rolling  as  to  give  the  epi-  or  peri -cycloid, 
requires  a  construction  similar  at  every  step  to  that  of  the  last  article.  The  same  remark  applies 
equally  to  the  path  of  a  point  within  or  beyond  the  circumference  of  the  rolling  circle.  This  is 
shown  in  Fig.  104,  before  describing  which  in  detail,  however,  we  will  summarize  the  steps  for  any 
and  all  trochoids. 

Letting  P  represent  the  tracing  point,  R  the  centre  of  the  rolling  circle  and  F  that  of  the  fixed 
circle,  we  draw  (1)  a  circle  through  P,  centre  R;  (2)  a  circle  (path  of  centres)  through  R,  centre 
F;  (3)  ascertain  by  a  proportion  (as  described  in  the  last  article)  how  many  degrees  of  arc  on 
either  circle  are  equal  to  the  prescribed  arc  of  contact  on  the  other;  (4)  on  the  path  of  centres  lay 


EPI-     AND    PERI-TROCHOIDS. 


61 


±O-i. 


off — from  the  initial  position  of  R  and  in  the  direction  of  intended  rolling — whatever  number  of 
degrees  of  contact  has  been  assigned  or  ascertained  for  the  fixed  circle,  and  divide  this  arc  by  radii 
from  F  into  any  number  of  equal  parts,  to  obtain  the  successive  positions  of  R,  as  Rl}  R.2,  etc.; 
(5)  on  the  circle  through  P  lay  off — from  the  initial  position  of  P,  and  in  the  direction  in  which 
it  will  move  when  the  assigned  rolling  occurs — the  same  number  of  degrees  that  have  been  assigned 
or  calculated  as  the  contact  arc  of  the  rolling  circle,  and  divide  such  arc'  into  the  same  number  of 
equal  parts  that  was  adopted  for  the  division  of  the  path  of  centres;  (6)  through  the  points  of 
division  obtained  in  the  last  step  draw  "circles  of  distance"  with  centre  F,  numbering  them  from 
P;  (7)  finally,  to  get  the  suc- 
cessive positions  of  P,  use  R  P 
(the  "  tracing  radius ")  as  a  con- 
stant radius,  and  cut  each  circle 
of  distance  by  an  arc  from  the 
like  -  numbered  position  from  R, 
selecting,  of  course,  the  right  one 
of  the  two  points  in  which  said 
curves  will  always  intersect  when 
not  tangent. 

In  Fig.  104  the  path  of  the 
point  P  is  determined  (a)  as  car- 
ried by  the  circle  called  "  first 
generator,"  rolling  on  the  exterior 
of  the  "first  director";  (b)  as 
carried  by  the  "second  generator" 
which  rolls  on  the  exterior  of  the 
"second  director" — which  it  also 
encloses.  In  the  first  case  the 
resulting  curve  is  a  prolate  epi- 
trochoid;  in  the  second  a  curtate 
peritrochoid;  but  such  values  were 
taken  for  the  diameters  of  the 
circles,  that  P  traced  the  same 
curve  under  either  condition  of 
rolling.*  These  (before  reduction 
with  the  camera)  were  3"  and  2" 
for  first  generator  and  first  director, 
respectively. 

For  the  epitrochoid  a  semi -circle  is  drawn  through  P  from  rolling  centre  R;  similarly  with 
centre  p  for  the  peritrochoid.  Dividing  these  semi -circles  into  the  same  number  of  equal  parts,  draw 
next  the  dotted  "  circles  of  distance "  through  these  points,  all  from  centre  F.  •  The  figure  illustrates 
the  special  case  where  the  two  sets  of  "  circles  of  distance "  coincide.  The  various  positions  of 
P,  as  Pn  P2,  etc.,  are  then  located  by  arcs  of  radii  RP  or  p  P,  struck  from  the  successive  positions 
of  R  or  p  and  intersecting  the  proper  "circle  of  distance." 


•Regarding  their  double  generation  refer  to  the  Appendix.    In  illustrating  both  methods  In  one  figure  it  will  add  greatly 
to  the  appearance  and  also  the  intelligibility  of  the  drawing  if  colors  are  used,  red  for  one  construction  and  blue  for  the  other. 


62 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


For   example,   the    turning    of  P   through    the    angle  PR  I    about  R    would    bring  P  somewhere    upon 

the   circle   of  distance  through   point   1;    but   that   amount   of   turning   would   be    due  to   the  rolling  of 

the   first    generator   over  the    arc   m  Q,   which    would    bring  n    upon   Q    and    carry  R  to  7?,;    P   would 

therefore    be    at  /',,    at    a    distance  RP   from  Rlt    and    on    the    dotted    arc    through  1.      Similarly  in 
relation   to   p.      When   s  reached   k,   in   the   rolling,   we   would   find  P  at  P2. 

Each   position   of  P  is  joined   with   each   of  the   centres   from   which   it   could   be  obtained. 

SPECIAL    TROCHOIDS. 

180.  The    Ellipse    and   Straight    Line.      Two    circles    are    called    Cardanic*    if   tangent    internally  and 
the   diameter   of  one   is   twice  that  of  the   other.      If  the  smaller    roll    in  the   larger,   all   points  in    the 
plane    of   the    generator    will    describe    ellipses    except    points    on  the   circumference,   each  of   which  will 
move  in  a   straight  line  —  a  diameter  of  the  director.      Upon  this  latter  property  the  mechanism  known 
as    "White's    Parallel    Motion"    is    based,    in    which     a    piston-rod    is    pivoted   to    a    small    gear-wheel 
which   rolls   on   the   interior   of   a   toothed   annular   wheel   whose   diameter  is    twice    that    of  the  pinion. 

181.  The  Limacon  and   Cardioid.      The    Limacon    is    a    curve    whose    points    may   be    obtained    by 
drawing   random   secants  through   a   point   on    the    circumference   of 

a  circle,  and  on  each  laying  off  a  constant  distance,  on  each  side 
of  the  second  point  in  which  the  secant  cuts  the  circle. 

In  Fig.  105  let  0  v  and  0  d  be  random  secants  of  the  circle 
Ons;  then  if  nv,  np,  ca  and  cd  are  each  equal  to  some  con- 
stant, b,  we  shall  have  v,  p,  a  and  d  as  four  points  of  a  Lima9on. 
Refer  points  on  the  same  secant,  as  a  and  d,  to  0  and  the  diam- 
eter Os;  we  then  have  Od  — p=  Oc+cd=2r cos6  +  b,  while  Oa  = 
2  r  cos  6 — b ;  hence  the  polar  equation  is  p  =  2rcos#±i. 

When  b  =  L2r  the  Limacon  becomes  a  Cardioid.^    (See  Fig.  106). 

182.  All    Limafons,    general    and    special,    may    be    generated    either    as     epi-    or    peri-trochoidal 
curves:    as    gpi-trochoids    the    generator    and    director    must    have    equal    diameters,    any    point    on    the 
circumference    of   the    generator    then    tracing    a    Cardioid,   while    any   point    on    the   radius    (or    radius 
produced)   describes   a  Limacon;    as  j>m-trochoids  the  larger  of  a   pair  of  Cardanic  circles  must  roll  on 
the  smaller,   the   Cardioid   and   Limacon   then   resulting,   as   before,   from    the    motion    of   points    respec- 
tively  on  the  circumference   of  the  generator,   or  within  or  without  it. 

183.  In   Fig.   106   the   Cardioid   is   obtained  as   an   epicycloid,  being   traced  by   point  P  during   one 
revolution   of  the   generator  PHm  about   an   equal   directing   circle  msO. 

As    a    Limacon  we    may  get   points   of  the   Cardioid,   as   y   and  z,   by   drawing   a   secant  through   0 
and  laying   off  s  y  and  s  z   each   equal   to   2  r. 

184.  The  Limayon  as  a    Trisectrix.      Three    famous   problems    of  the  ancients    were    the  squaring   of 
the   circle,  the   duplication   of  the   cube   and   the   trisection   of  an   angle.      Among  the   interesting  curves 
invented  by   early   mathematicians    for    the    purpose    of   solving    one    or    the    other    of   these  problems, 
were    the   Quadratrix    and    Conchoid,   whose    construction    is    given    later    in    this    chapter;    but    it    has 
been   found  that  certain   trochoids   may   as   readily   be   employed   for  trisection,   among  them   the   Lima- 
gon   of  Fig.    106,   frequently   called   the  Epitrochoidal    Trisectrix. 

When   constructed   as   a   Limacon    we    find    points   as   G  and    X,    on    any   secant  R  X  of  the    circle 
called   "  path    of  centres,"   by   making   S  X  and   S  G  each   equal   to   the   radius   of  that   circle. 


*Tenn    due    to   Keuleaux,    and   based    upon    the    fact  that    Cardano  (16th  century)   was  probably  the  first  to    investigate   the 
paths  described  by  points  during  their  rolling.  fFrom   Cardis,  the  Latin  for  heart. 


SPECIAL    TROCHOIDS. 


63 


185.  To  trisect  an  angle,  as  M  R  F,  l>y  means  of  this  epitrochoid,  bisect  one  side  of  the  angle,  as 
F  R,  at  m;  use  ?7i  R  and  mF  as  radii  for  generator  and  director  respectively  of  an  epitrochoid  hav- 
ing a  tracing  radius,  RF,  equal  to  twice  that  of  the  generator.  Make  RN=RF  and  draw  NF;  this 
will  cut  the  Lima9on  FT}RQ  (traced  by  point  F  as  carried  by  the  given  generator)  in  a  point  1\  . 
The  angle  T^F-  will  then  be  one  -third  of  NRF,  which  may  be  proved  as  follows:  F  reaches  T, 
by  the  rolling  of  arc  mn  on  arc  win,.  These  arcs  are  subtended  by  equal  angles,  <#>,  the  circles  being 
equal.  During  this  rolling  R  reaches  .R,,  bringing  R  F  to  R  1  7\  .  In  the  triangles  T^R^F  and  RFRt 
the  side  FRl  is  common,  angles  <j>  equal,  and  side  -fi^T7,  equal  to  side  RF;  the  line  R  Tl  is  there- 
fore parallel  to  RtF,  whence  angle  T^RF  must  also  equal  <f>.  In  the  triangle  RFRl  we  denote  by  0 


the    angles    opposite    the    equal  sides  RF  and  R1F;   then 


or 


1  QQO  _   i 

0=-  —  „  —  ™.      In    triangle 


.  10S. 


we  have  the  angle  at  F  equal  to  0  — <£,  and  2  (6  —  <£)  +  *  +  <^  =  180°,  which  gives  a;=2<£,  by 
substituting  the  value  of  6  from  the  previous  equation. 

186.  The  Involute.  As  the  opposite  extreme  of  a  circle  rolling  on  a  straight  line  we  may  have 
the  latter  rolling  on  a  circle.  In  this  case  the  rolling  circle  has  an  infinite  radius.  A  point  on  the 
straight  line  describes  a  curve  called  the  involute.  This  would  be  the  path  of  the  end  of  a  thread 
if  the  latter  were  in  tension  while  being  unwound  from  a  spool. 

In  Fig.  107  a  rule  is  shown,  tangent  at  M  to  a  circle  on  which  it  is  supposed  to  roll.  Were  a 
pencil -point  inserted  in  the  centre  of  the  circle  at  j  (which  is  on  the  line  ux  produced)  it  would 
trace  the  involute.  When  j  reaches  a,  the  rule  will  have  had  rolling  contact  with  the  base  circle 
over  an  arc  uts  —  u  whose  length  equals  line  uxj.  Were  a  the  initial  point,  we  would  obtain  6,  c, 


64 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


etc.,  by  making  tangent  mb  =  arc  ma;  tangent  nc=arc  na.  Each  tangent  thus  equals  the  arc  from 
the  initial  point  to  the  point  of  tangency. 

187.  The  circle  from  which  the  involute  is  derived  or  evolved  is  called  the  evolute.  Were  a 
hexagon  or  other  figure  to  be  taken  as  an  evolute,  a  corresponding  involute  could  be  derived;  but 
the  name  "involute,"  unqualified,  is  understood  to  be  that  obtained  from  a  circle. 

From  the  law  of  formation  of  the  involute,  the  rolling  line  is  in  all  its  positions  a  normal  to 
the  curve;  the  point  of  tangency  on  the  evolute  is  an  instantaneous  centre,  and  a  tangent  at  any 
point,  as  /,  is  a  perpendicular  to  the  tangent,  fq,  from  /  to  the  base  circle. 

Like  the  cycloid,  the  involute  is  a  correct  working  outline  for  the  teeth  of  gear-wheels;  and 
gears  manufactured  on  the  involute  system  are  to  a  considerable  degree  supplanting  other  forms. 

A  surface  known  as  the  developable  helicoid  (see  Figs.  209   and  270)   is   formed    by  moving  a  line 

--  1O7. 


so  as  to  be  always  tangent  to  a  given  helix.  It  is  interesting  in  this  connection  to  notice  that  any 
plane  perpendicular  to  the  axis  of  the  helix  would  cut  such  a  surface  in  a  pair  of  involutes.* 

188.  The  Spiral  of  Archimedes.  This  curve  is  generated  by  a  point  having  a  uniform  motion 
around  a  fixed  point — the  pole — combined  with  uniform  motion  toward  or  from  it. 

In  Fig.  107,  with  0  as  the  pole,  if  the  angles  0  are  equal,  and  0  D,  OE  and  Oy3  are  in  arith- 
metical progression,  then  the  points  D,  E  and  i/3  are  points  of  an  Archimedean  Spiral. 

This  spiral  can  be  trochoidally  generated,  simultaneously  with  the  involute,  by  inserting  a  pencil 
point  at  y  in  a  piece  carried  by — and  at  right  angles  with — the  rule,  the  point  y  being  at  a  distance, 


•The  day  of  writing  the  above  article  the  following  item  appeared  in  the  New  York  Evening  Post:  "Visitors  to  the  Royal 
Observatory,  Greenwich,  will  hereafter  miss  the  great  cylindrical  structure  which  has  for  a  quarter  century  and  more  covered 
the  largest  telescope  possessed  by  the  Observatory.  Notwithstanding  its  size  the  Astronomer  Koyal  has  now  procured  through 
the  Lords  Commissioners  a  telescope  more  than  twice  as  large  as  the  old  one. . . .  The  optical  peculiarities  embodied  in  the 
new  instrument  will  render  it  one  of  the  three  most  powerful  telescopes  at  present  in  existence....  The  peculiar  architectural 
feature  of  the  building  which  is  to  shelter  the  new  telescope  is  that  its  dome,  of  thirty-six  feet  diameter,  will  surmount  a 
tower  having  a  diameter  of  only  thirty-one  feet.  Technically,  the  form  adopted  is  the  surface  generated  by  the  revolution  of 
an  involute  of  a  circte." 


SPECIAL    TROCHOIDS. 


65 


1OB. 


xy,  from  the  contact -edge  of  the  rule,  equal  to  the  radius  Os  of  the  base  circle  of  the  involute;  for 
after  the  rolling  of  ux  over  an  arc  ut  we  shall  have  txl  as  the  portion  of  the  rolling  line  between 
x  and  the  point  of  tangency,  and  xy  will  have  reached  xlyl.  If  the  rolling  be  continued  y  will 
evidently  reach  0.  We  see  that  Oy  =  ux,  and  Oyl  =  txl;  but  the  lengths  ux  and  txl  are  propor- 
tional to  the  angular  movement  of  the  rolling  line  about  0,  and  as  the  spiral  may  be  defined  as 
that  curve  in  which  the  length  of  a  radius  vector  is  directly  proportional  to  the  angle  through  which 
it  has  turned  about  the  pole,  the  various  positions  of  y  are  evidently  points  of  such  a  curve. 

189.  A    Tangent  to  the  Spiral  of  Archimedes.     Were   the    pole,   0,  given,   and    a  portion   only   of  the 
spiral,   we    could    draw    a    tangent  at    any   point,   y , ,   by   determining     the    circle    on    which   the    spiral 
could  be  trochoidally  generated,  then  the  instantaneous  centre  for  the  given    position    of   the    tracing- 
point,  whence  the   normal   and    tangent   would   be   derived  in   the  usual  way.      The  radius   Ot  of  the 
base  circle  would  equal  wy — the  difference  between  two  radii  vectores  Oy  and  Oz  which  include    an 
angle  of  57°  29+,   (the  angle  which  at  the  centre  of  a  circle    subtends  an  arc  equal    to  the  radius). 
The  instantaneous   centre,   t,  would  be  the  extremity   of  that  radius  which  was  perpendicular  to  Oyl. 
The   normal  would  be   tyr,  and  the   tangent   TTl  perpendicular   to   it. 

190.  The  spiral  of  Archimedes  is  the   right  section  of  an   oblique  helicoid.     (Art.  357).    It  is  also 
the    proper    outline   for  a  cam  to  convert  uni- 
form    rotary    into    uniform    rectilinear    motion, 

and  when  combined  with  an  equal  and  oppo- 
site spiral  gives  the  well-known  form  called 
the  heart -cam.  As  usually  constructed  the  act- 
ing curve  is  not  the  true  spiral,  but  a  curve 
whose  points  are  at  a  constant  distance  from 
the  theoretical  outline  equal  to  the  radius  of 
the  friction -roller  which  is  on  the  end  of  the 
piece  to  be  raised.  Qs2  (Fig.  107)  is  a  small 
portion  of  such  a  "parallel  curve." 

191.  If   a    point   travel   on    the  surface  of 
a    cone    so    as    to    combine    a    uniform    motion 
around  the  axis  with  a  uniform  motion  toward 
the  vertex  it  will   trace   a    conical    helix,  whose 
orthographic    projection    on    the    plane    of  the 
base  will   be  a  spiral  of  Archimedes. 

.  In  Fig.  108  a  top  and  front  view  are 
given  of  a  cone  and  helix.  The  shaded  por- 
tion is  the  development  of  the  cone,  that  is, 
the  area  equal  to  the  convex  surface,  and 
which  —  if  rolled  up  —  would  form  the  cone. 
To  obtain  the  development  draw  an  arc 
A'G"A"  of  radius  equal  to  an  element.  The 
convex  surface  of  the  cone  will  then  be  repre- 
sented by  the  sector  A'O'A",  whose  angle  6 
may  be  found  by  the  proportion  OA:  ff  A':: 
circumference  of  the  cone's  base. 

The  student  can  make  a  paper  model  of  the  cone  and  helix  by  cutting  out  a  sector  of  a  circle, 


61:360°,   since  the   arc  A'G"A"   must 


the  entire 


66  THEORETICAL    AND    PRACTICAL     GRAPHICS. 

making   allowance   for  an   overlap   on   which   to    put   the   mucilage,   as    shown    by   the    dotted    lines   O'y 
and  y  v  z  in   the   figure. 

The   development  of  a   conical   helix   is   the   same   kind   of  spiral   as   its   orthographic   projection. 

PARALLEL    CURVES. 

192.  A  parallel  curve  is  one  whose  points  are  at  a  constant  normal  distance  'from  some  other 
curve.  Parallel  curves  have  not  the  same  mathematical  properties  as  those  from  which  they  are 
derived,  except  in  the  case  of  a  circle;  this  can  readily  be  seen  from  the  cam  figure  under  the  last 
heading,  in  which  a  point,  as  «,,  of  the  true  spiral,  is  located  on  a  line  from  0  which  is  by  no 
means  in  the  direction  of  the  normal  to  the  curve  at  «,,  upon  which  lies  the  point  s2  of  the 
parallel  curve. 


Instead  of  actually  determining  the  normals  to  a  curve  and  on  each  laying  off  a  constant 
distance,  we  may  draw  many  arcs  of  constant  radius,  having  their  centres  on  the  original  curve; 
the  desired  parallel  will  be  tangent  to  all  these  arcs. 

In  strictly  mathematical  language  a  parallel  curve  is  the  envelope  of  a  circle  of  constant  radius 
whose  centre  is  on  the  original  curve.  We  may  also  define  it  as  the  locus  of  consecutive  inter- 
sections of  a  system  of  equal  circles  having  their  centres  on  the  original  curve. 

If  on  the  convex  side  of  the  original  the  parallel  will  resemble  it  in  form,  but  if  within,  the 
two  may  be  totally  dissimilar.  This  is  well  illustrated  in  Fig.  109,  in  which  the  parallel  to  a 
Lemniscate  is  shown. 

The  student  will  obtain  some  interesting  results  by  constructing  the  parallels  to  ellipses,  parabolas 
and  other  plane  curves. 

THE    CONCHOID    OF    NICOMEDES. 

193.  The  Conchoid,  named  after  the  Greek  word  for  shell*  may  be  obtained  by  laying  off  a  con- 
stant length  on  each  side  of  a  given  line  M ' N  (the  directrix),  upon  radials  through  a  fixed  point  or 
pole,  0  (Fig.  110).  If  imv  =  mn=8x  then  v,  n  and  x  are  points  of  the  curve.  Denote  by  a  the 
distance  of  0  from  MN,  and  use  c  for  the  constant  length  to  be  laid  off;  then  if  a<c  there  will 
be  a  loop  in  that  branch  of  the  curve  which  is  nearest  the  pole, — the  inferior  branch.  If  a  =  c  the 
curve  has  a  point  or  cusp  at  the  pole.  When  «>c  the  curve  has  an  undulation  cr  wave- form 
towards  the  pole. 

*A  scries  of  curves  much  more  closely  resembling  those  of  a  shell  can  be  obtained  by  tracing  the  paths  of  points  on  the 
piston-rod  of  an  oscillating  cylinder.  See  Arts.  157  and  158  for  the  principles  of  their  construction. 


THE    CONCHOID.— THE    QUADRATRIX. 


67 


Ov  —  c+Om;   On=c — Om;   we  may  therefore   express    the  relation  to   0  of  points   on  the  curve 
by   the   equation  p  =  c±  0  m=c±asec<f>. 


rig-,  no. 


194.  Mention  has  already  heen  made    (Art.   184)    of   the    fact    that    this    was    one    of   the    curves 
invented    in    part    for    the    purpose    of   solving   the   problem   of   the   trisection  of  an  angle.      Were   m  0  x 
(or   <£)   the   angle   to   be  trisected   we  would   first   draw  pqr,  the   superior  branch   of  a   conchoid   having 
the   constant,   c,   equal   to   twice   Om.      A   parallel   from   m   to   the    axis    will    intersect    the    curve    at    q; 
the   angle   pOq  will    then    be    one-third    of   <£:    for    since    bq=20m    we    have    mg  =  2  Omcos/3;    also 
mq  :  Om: :  sin  6:  sin/3;     hence     2  Omcos/3:  Om  : :  sin 6  :  si?i/3,      whence     sin  6  =  2  sin  f3  cos  (3=  sin  2  (3     (from 
known    trigonometric    relations).      The    angle    6    is   therefore   equal   to   twice  /3,   which    makes   the   latter 
one -third   of  angle   <f>. 

195.  To   draw   a  tangent  and    normal    at    any   point    v,  we    find    the    instantaneous    centre  o   on  the 
principle   that  it   is   at  the   intersection   of    normals   to   the   paths   of  two  moving  points   of   a   line,   the 
distance   between   said   points   remaining    constant.      In    tracing   the   curve,  the   motion   of  0   (on   Ov)  is 
— at   the   instant   considered — in   the    direction   Ov;   Oo  is   therefore  the   normal.     The    point  m   of  Ov 
is   at  the   same   moment  moving  along  M N,  for   which  mo  is  the  normal.      Their  intersection  o  is  then 
the    instantaneous    centre,   and   o  v  the   normal   to    the    conchoid,   with   v  z    perpendicular    to   o  v   for  the 
desired   tangent. 

196.  This   interesting   curve  may   be   obtained   as   a   plane   section   of  one   of  the   higher  mathemat- 
ical  surfaces.     If -two   non- intersecting  lines  —  one   vertical,    the    other    horizontal — be    taken    as    guiding 
lines   or   directrices  of  the   motion   of  a   third   straight   line   whose   inclination  to  a  horizontal  plane  is  to 
be   constant,   then   horizontal  planes   will    cut    conchoids    from    the    surface    thus    generated,   while    every 
plane    parallel    to    the    directrices    will    cut    hyperbolas.       From    the    nature    of    its    plane    sections    this 
surface   is   called  the    Conchoidal  Hyperboloid.      (See   Fig.   219). 

THE     QUADRATRIX     OF    DINOSTRATUS. 

197.  In  Fig.   Ill   let  the    radius   0  T  rotate    uniformly  about  the  centre;    simultaneously  with  its 
movement    let  M  N   have    a    uniform    motion    parallel    to    itself,   reaching  A  B    at    the    same  time  with 
radius   0  T;    the    locus    of   the    intersection    of  M  N   with    the    radius    will    be    the  Quadratrix.      Points 


68 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


exterior  to  the  circle  may  be  found  by  prolonging  the  radii  while  moving  MN  away  from  A  B. 
As  the  intersection  of  M N  with  OB  is  at  infinity,  the  former  becomes  an  asymptote  to  the  curve 
as  often  as  it  moves  from  the  centre  an  additional  amount  equal  to  the  diameter  of  the  circle; 
the  number  of  branches  of  the  Quadratrix  may  therefore 
be  infinite.  It  may  be  proved  analytically  that  the  curve 
crosses  0  A  at  a  distance  from  0  equal  to  2  r  H-  IT. 

198.  To    trisect    an    angle,    as    T  0  a,    by    means   of   the 
Quadratrix,   draw    the    ordinate   ap,  trisect  p  T  by   s  and  x 
and     draw     8  c    and    %  m;     radii    0  c    and     0  m    will    then 
divide    the    angle    as     desired:     for    by     the     conditions     of 
generation    of   the    curve    the    line   MN    takes     three    equi- 
distant  parallel   positions    while    the    radius    describes    three 
equal   angles. 

THE     CISSOID     OF     DIOCLES. 

199.  This  curve  was   devised  for  the  purpose  of   obtaining  two    mean    proportionals    between  two 
given   quantities,   by   means   of  which   the   duplication   of  the   cube   might  be   effected. 

The  name  was  suggested  by  the  Greek  word  for  ivy,  since  "the  curve  appears  to  mount  along 
its  asymptote  in  the  same  manner  as  that  parasite  plant  climbs  on  the  tall  trunk  of  the  pine."* 

This  was  one  of  the  first  curves  invented  after  the  discovery  of  the  conic  sections.  Let  C  (Fig. 
112)  be  the  centre  of  a  circle,  ACE  a  right  angle,  NS  and  MT  any  pair  of  ordinates  parallel  to 


and  equidistant  from  CE;  then  a  secant  from  A  through  the  extremity  of  either  ordinate  will  meet 
the  other  ordinate  in  a  point  of  the  cissoid.  A  T  and  NS  give  P;  A  S  and  M  T  give  Q. 

The  tangent  to   the   circle   at  B  will   be   an   asymptote   to   the   curve. 

It  is  a  somewhat  interesting  coincidence  that  the  area  between  the  cissoid  and  its  asymptote  is 
the  same  as  that  between  a  cycloid  and  its  base,  viz.,  three  times  that  of  the  circle  from  which 
it  is  derived. 

200.  Sir  Isaac  Newton  devised  the  following  method  of  obtaining  a  cissoid  by  continuous  motion: 
Make  AV=AC;  then  move  a  right-angled  triangle,  of  base  =  V (7,  so  that  the  vertex  F  travels  along 


"Leslie.     Geometrical  Analysis.     1821. 


THE    CISSOID.— THE    TRACTRIX.  69 

the  line  DE  while  the  edge  JK  always  passes  through  V;  then  the  middle  point,  L,  of  the  base  FJ, 
will  trace  a  cissoid.  This  construction  enables  us  readily  to  get  the  instantaneous  centre  and  a  tangent 
and  normal;  for  FH  is  normal  to  FC — the  path  of  F,  while  nV  is  normal  to  the  motion  of  /  toward 
/  V;  the  instantaneous  centre  n  is  therefore  at  the  intersection  of  these  normals.  For  any  other 
point  as  P  we  apply  the  same  principle  thus :  With  radius  A  C  and  centre  P  obtain  x;  draw  Px, 
then  Vz  parallel  to  it;  a  vertical  from  x  will  meet  Vz  at  the  instantaneous  centre  y,  whence  the 
normal  and  tangent  result  in  the  usual  way.  The  point  y  does  not  necessarily  fall  on  nV. 

Since  nV  and  FJ  are  perpendicular  to  J  V  they  are  parallel.  So  also  must  Vz  be  parallel  to 
Px,  regardless  of  where  P  is  taken. 

201.  Two  quantities  m  and  n  will  be  mean  proportionals  between  two  other  quantities  a  and  b 
if  m*=na  and  n'*=mb;  that  is,  if  m3=«26  and  if  n*=<ib2. 

If  6  =  2  a  we  will  find,  from  the  relation  m'4  =  a'2b,  that  m  will  be  the  edge  of  a  cube  whose 
volume  equals  2  a3. 

To  get  two  mean  proportionals  between  quantities,  r  and  b,  make  the  smaller,  r,  the  radius  of  a 
circle  from  which  derive  a  cissoid.  Were  APR  the  derived  curve  we  would  then  make  Ct  equal 
to  the  second  quantity,  b,  and  draw  B  t,  cutting  the  cissoid  at  Q.  A  line  A  Q  would  cut  off  on 
Ct  a  distance  Cv  equal  to  m,  one  of  the  desired  proportionals;  for  m3  will  then  equal  r'!b,  as  may 
be  thus  shown  by  means  of  similar  triangles: 

r3  MO3 
Cv:  MQ::CA:  MA     whence      Cv*  =  '  I,.?        

MA 


Ct:MQ::CB:BM  C«=:-Dnrr 

B  M 


M  Q-.MA::  S X :  A  N : :  /  A  N.  BN:A  N,   whence   MQ  =MA^AX.BN 

From   (2)    we   have   J/Q  =  —  (4) 


r 


(V    K       «  MA*(AN.BN) 

W  Mty  =—         .  m —      - (o; 


Replacing  M  Q3  in   equation    (1)    by   the    product    of   the    second    members    of    equations    (4)    and    (5) 
gives    Cv3   (i.e.,   ms)  =  r26. 

By  interchanging  r  and  b  we  obtain  n,  the  other  mean  proportional;  or  it  might  be  obtained 
by  constructing  similar  triangles  having  r,  b  and  m  for  sides. 

THE     TRACTRIX. 

202.  The  Tractrix  is  the  involute  of  the  curve  called  the  Catenary  (Art.  214)  yet  its  usual  con- 
struction is  based  on  the  fact  that  if  a  series  of  tangents  be  drawn  to  the  curve,  the  portions  of 
such  tangents  between  the  points  of  tangency  and  a  given  line  will  be  of  the  same  length ;  or,  in 
other  words,  the  intercept  on  the  tangent,  between  the  directrix  and  the  curve,  will  be  constant.  A 
practical  and  very  close  approximation  to  the  theoretical  curve  is  obtained  by  taking  a  radius  Q  R 
(Fig.  11.3)  and  with  a  centre  a,  a  short  distance  from  R  on  Q  R,  obtaining  b,  'which  is  then  joined 
with  a.  On  a  b  a  centre  c  is  similarly  taken  for  another  arc  of  the  same  radius,  whence  c  d  is 
obtained.  A  sufficient  repetition  of  this  process  will  indicate  the  curve  by  its  enveloping  tangents, 
or  a  curve  may  actually  be  drawn  tangent  to  all  these  lines.  Could  we  take  «,  6,  c,  etc.,  as 
mathematically  consecutive  points  the  curve  would  be  theoretically  exact.  The  line  QS  is  an  asymp- 
tote to  the  curve. 


70 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


The  area  between  the  completed    branch  RPS   and    the    lines  QR    and  QS   would    be    equal   to 
a    quadrant    of   the    circle    on    radius   Q  R. 

_2 £i_o         203.     The    surface    generated    by    revolving    the    trac- 


trix  about  its  asymptote  has  been  employed  for  the 
foot  of  a  vertical  spindle  or  shaft,  '  and  is  known  as 
Schiele's  Anti- Friction  Pivot.  The  step  for  such  a  pivot 
is  shown  in  sectional  view  in  the  left  half  of  the  figure. 
Theoretically,  the  amount  of  work  done  in  overcoming 
friction  is  the  same  on  all  equal  areas  of  this  surface. 
In  the  case  of  a  bearing  of  the  usual  kind,  for  a 
cylindrical  spindle,  although  the  pressure  on  each  square 
inch  of  surface  would  be  constant,  yet,  as  unit  areas  at 
different  distances  from  the  centre  would  pass  over  very 
different  amounts  of  space  in  one  revolution,  the  wear 
upon  them  would  be  necessarily  unequal.  The  rationale  of 
the  tractrix  form  will  become  evident  from  the  following 
consideration:  If  about  to  split  a  log,  and  having  a 
choice  of  wedges,  any  boy  would  choose  a  thin  one  rather  than  one  with  a  large  angle,  although 
he  might  not  be  able  to  prove  by  graphical  statics  the  exact  amount  of  advantage  the  one  would 
have  over  the  other.  The  theory  is  very  simple,  how- 
ever, and  the  student  may  profitably  be  introduced  to  it. 
Suppose  a  ball,  c,  (Fig.  114)  struck  at  the  same  instant 
by  two  others,  a  and  b,  moving  at  rates  of  six  and  eight 
feet  a  second  respectively.  On  a  c  and  b  c  prolonged  take 
c  e  and  c  h  equal,  respectively,  to  six  and  eight  units  of 
some  scale;  complete  the  parallogram  having  these  lines 

as  sides;  then  it  is  a  well-known  principle  in  mechanics*  that  erf  — the  diagonal  of  this  parallel- 
ogram —  will  not  only  represent  the  direction  in  which  the  ball  c  will  move,  but  also  the  distance  — 
in  feet,  to  the  scale  chosen  —  it  will  travel  in  one  second.  Evidently,  then,  to  balance  the  effect  of 
balls  a  and  b  upon  c,  a  fourth  would  be  necessary,  moving  from  d  toward  c  and  traversing  dc  in 
the  same  second  that  a  and  b  travel,  so  that  impact  of  all  would  occur  simultaneously.  These 
forces  would  be  represented  in  direction  and  magnitude  (to  some  scale)  by  the  shaded  triangle 
c'd'e',  which  illustrates  the  very  important  theorem  that  if  the  three  sides  of  a  triangle  —  taken  like 
c'e',  e'd',  d'c',  in  such  order  as  to  bring  one  back  to  the  initial,  vertex  mentioned  —  represent  in 
magnitude  and  direction  three  forces  acting  on  one  point,  then  these  forces  are  balanced. 

Constructing  now  a  triangle  of  forces  for  a  broad  and  thin 
wedge,  (Fig.  115)  and  denoting  the  force  of  the  supposed  equal  blows 
by  F  in  each  triangle,  we  see  that  the  pressures  are  greater  for  the 
thin  wedge  than  for  the  other;  that  is,  the  less  the  inclination  to 
the  vertical  the  greater  the  pressure.  A  pivot  so  shaped  that  as 
the  pressure  between  it  and  its  step  increased  the  area  to  be  traversed 
diminished  would  therefore,  theoretically,  be  the  ideal;  and  the  rate  of 
change  of  curvature  of  the  tractrix,  as  its  generating  point  approaches 
the  axis,  makes  it,  obviously,  the  correct  form. 


rig-,  us. 


•For  a  demonstration  the  student  may  refer  to  Hankine's  Applied  Mechanics,  Art.  51. 


THE    TRACTR IX  — WITCH    OF    AGNESI.-CARTESIAN    OVALS. 


71 


204.  Navigator's   charts   are   usually   made   by  Mercator's  projection   (so-called,   not  being  a  projection 
in    the    ordinary    sense,    but    with    the    extended    signification    alluded    to    in    the    remark    in    Art.    2). 
Maps   thus   constructed   have   this    advantageous    feature,   that  rhumb   lines  or   loxodromics  —  the   curves   on 
a   sphere  that   cut    all   meridians   at  the   same   angle  —  are   represented   as   straight  lines,  which   can   only 
be  the   case   if  the   meridians   are   indicated   by  parallel    lines.      The    law    of    convergence    of   meridians 
on   a   sphere  is,   that  the  length   of  a   degree   of    longitude   at   any   latitude   equals   that   of  a   degree  on 
the   equator  multiplied   by   the   cosine   (see   foot-note,   p.    31)   of   the    latitude;    when   the  meridians  are 
made    non- convergent    it    is,   therefore,   manifestly   necessary   that    the    distance    apart   of    originally   equi- 
distant   parallels    of   latitude    must    increase    at    the    same    rate;    or,   otherwise   stated,   as   on   Mercator's 
chart    degrees    of    longitude    are   all    made   equal,   regardless   of   the   latitude,   the   constant  length   repre- 
sentative  of  such    degree   bears   a   varying   ratio   to   the    actual    arc    on    the    sphere,   being    greater    with 
the    increase    in    latitude;    but    the    greater    the    latitude    the    less   its   cosine   or  the  greater  its   secant; 
hence    lengths    representative    of    degrees    of    latitude    will    increase    with    the    secant    of   the    latitude. 
Tables   have   been   constructed  giving   the   increments   of   the    secant    for    each    minute    of   latitude;    but 
it    is    an    interesting    fact    that    they    may   be    derived    from    the    Tractrix    thus:     Draw    a    circle    with 
radius   Q  R,  centre   Q   (Fig.    113);    estimate   latitude   on   such   circle    from  R    upward;    the    intercept    on 
QS   between    consecutive    tangents    to    the    Tractrix    will    be    the    increment    for    the    arc    of    latitude 
included  between   parallels   to   QS,   drawn   through   the   points   of  contact   of  said   pair   of  tangents." 

On   map   construction   the   student   is   referred   to   Chapter  XII,   or  to    Craig's    Treatise  on  Projections. 

THE     WITCH     OF     AGNESI. 

205.  If    on    any   line  S  Q,   perpendicular    to    the    diameter    of   a    circle,   a    point  S    be    so    located 
that  S  Q:  A  B  ::PQ:  Q B   then  .S   will    be    a    point    of   the    curve    called    the    Witch   of  Agnesi.      Such 
point    is    evidently    on    the    ordinate  P  Q    prolonged,    and    vertically    below    the    intersection    T  of   the 
tangent  at  A   by   the   secant   through  P. 


The  point  E,  at  the  same  level  as  the  centre   0,  is  a  diameter's  distance  from  the  latter. 

The  tangent  at  B  is  an  asymptote  to  the  curve. 

The  area  between  the  curve  and  its  asymptote  is  four  times  that  of  the  circle  involved  in  its 
construction. 

The  Witch,  also  called  the  Versiera,  was  devised  by  Donna  Maria  Gaetana  Agnesi,  a  brilliant 
Italian  lady  who  was  appointed  in  1750,  by  Pope  Benedict  XIV,  to  the  professorship  of  mathematics 
and  philosophy  in  the  University  of  Bologna. 

THE     CARTESIAN     OVAL. 

206.  This  curve,  also  called  simply  a  Cartesian,  after  its  investigator,  Descartes,  has  its  points 
connected  with  two  foci,  F'  and  F",  by  the  relation  m p'dtr.n p"  =k c,  in  which  c  is  the  distance 
between  the  foci,  while  m,  n  and  k  are  constant  factors. 


•Leslie.     Geometrical  Analysis.     Edinburgh,   1821. 


72 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


Salmon  states  that   we   owe  to   Chasles   the   proof  that   a  third   focus   may  be  found,  sustaining  the 

same    relation,    and    expressed    by    an    equation    of    similar    form.       (See 
Art.   209). 

The    Cartesian    is    symmetrical    with    respect    to    the    axis — the   line 
joining  the   foci. 

207.     To    construct    the    curve    from    the    first    equation   we   may   for 

convenience     write     mp'±np"  =  kc    in    the     form    p' ±  —  p"--     "--•    or 

by   denoting   -'-  by   b   and       "-   by   d,  it    takes  the   yet  more   simple   form 
p'±bp"  =  d.      Then   p"   will   have   two   values,  according   as   the   positive 
or    negative    sign    is    taken,    being    respectively    — 5—-   and    — , — ;    the    former    is    for    points    on    the 

inner    of   the    two    ovals    that    constitute    a    complete    Cartesian,   while    the    latter    gives    points    on    the 
outer   curve. 

To   obtain   p"  =  '^P-    take    F'    and    F"    (Fig. 


-.  ne. 


118)  as  foci;  F'S=d;  SK  at  some  random  acute 
angle  0  with  the  axis,  and  make  SH=^;  that 

is,  make  F' S:  SH: :  b  :  1.  Then  from  F'  draw 
an  arc  tfP,  of  radius  less  than  d,  and  cut  it  at 
P  by  an  arc  from  centre  F",  radius  S  T,  Tt  being 
a  parallel  to  F'  H;  then  P  is-  a  point  of  the 
inner  oval;  for  8t=d  —  p',  and  ST=p";  there- 
fore p":d  —  p' : :  .:  d,  whence  p"  -  — •:-*—. 

208.     If   an    arc   xyK  be   drawn   from   F' ,  with 
radius,  F'  x,  greater  than  d,  we   may  find  the  second 

value   of  p",   viz.,   p- — ,   by    drawing   xQ    parallel    to    F' H    to    meet   HS    prolonged;     for    QS    will 


equal    !— T— ,   in   which   p'  =  F'x.     Again    using  F"   as    a    centre,   and    a    radius  QS  =  p",  gives    points 

R  and  M  of  the   larger   oval. 

The    following    are    the    values    for    the    focal    radii    to    the    four   points    where   the    ovals    cut   the 
axes.      (See   Fig.   117). 

For  A,       p"  =  'L^=c  +  p'   whence  P'  =  F'A  =  d+b° 


B, 


d-P' 


d  —  b  c 


( I  +  b  c 

1  +b 

d  —  be 


b  "1—6 

The  construction  -  arcs   for  the   outer  oval   must  evidently  have   radii  between  the   values   of  p'  for  A 
and  B  above;    and   for  the   inner   oval   between  those   of   a  and   b. 


CA  R  TE  S I A  N    0  VA  L  S.  —  CA  US  TICS. 


73 


The    numerical    values    from    which     Fig.    118    was   constructed    were    m  =  3;    n=2;    c  =  l;    k  =  3. 

209.  The   Third  Focus.      Fig.    118   illustrates   a   special   case,   but,  in   general,  the   method   of  finding 
a  third   focus  F'"   (not    shown)   would    be    to    draw    a    random    secant  F'  r    through  F',   and    note    the 
points  P    and   G  in    which    it    cuts    the    ovals  —  these    to    be    taken    on    the    same   side   of  F' ',   as   two 
other   points   of  intersection   are   possible;    a   circle   through  P,   G  and  F"   would    cut    the    axis    in    the 
new   focus  sought.      Then  denoting  by   C  the  distance  F'  F'",  we  would   find  the  factors  of  the  original 
equation    appearing    in    a    new    order;    thus,    kp'±np'"  =  mC,    which — for    purposes    of    construction  — 
may   be   written   p'±b'p'"=d'. 

If  obtained  from  the  foci  F"  and  F'"  the  relation  would  be  m  p'"  —kp"-=  ±  n  C',  in  which  C' 
equals  F"  F"'.  Writing  this  in  the  form  p"'  —Bp"  =  ±D  we  have  the  following  interesting  cases: 
(a)  an  ellipse  for  D  positive  and  B  —  - 1 ;  (b)  an  hyperbola  for  D  positive  and  R  =  +  1 ;  (c)  a 
lima^on  for  D  =  C'  B ;  (d)  a  cardioid  for  B  —•-  +  1  and  Z)  =  C". 

210.  The    following    method    of    drawing    a    Cartesian    by   continuous   motion   was 
devised    by    Prof.    Hammond:      A    string    is    wound,    as    shown,   around    two    pulleys 
turning   on    a    common    axis;    a    pencil    at  P    holds    the    string    taut    around    smooth 
pegs    placed    at    random    at    Fl    and    F^;     if    the    wheels    be    turned    with    the    same 
angular  velocity,  and  the  pencil   does  not  slip   on  the  string,  it   will   trace   a   Cartesian 
having   Ft   and   F.f   as   foci.* 

If  the  pulleys  are  equal  the  Cartesian  will  become  an  ellipse;  if  both  threads 
are  wound  the  same  way  around  either  one  of  the  wheels  the  resulting  curve  will  be 
an  hyperbola. 

211.  It    is    a    well -known    fact    in    Optics   that  the   incident   and   reflected   ray   make   equal  angles 
with   the   normal   to   a  reflecting   surface.      If  the  latter  is   curved  then  each   reflected   ray  cuts  the   one 

next  to  it,  their  consecutive  intersections  giving  a  curve  called  a 
caustic  by  reflection.  Probably  all  have  occasionally  noticed  such  a 
curve  on  the  surface  of  the  milk  in  a  glass,  when  the  light  was 
properly  placed.  If  the  reflecting  curve  is  a  circle  the  caustic  is 
the  evolute  of  a  limagon. 

In  passing  from  one  medium  into  another,  as  from  air  into 
water,  the  deflection  which  a  ray  of  light  undergoes  is  called 
refraction,  and  for  the  same  media  the  ratio  of  the  sines  of  the 
angles  of  incidence  and  refraction  (0  and  <£,  Fig.  120)  is  constant. 
The  consecutive  intersections  of  refracted  rays  give  also  a  caustic, 
which,  for  a  circle,  is  the  evolute  of  a  Cartesian  Oval.  The  proof  of  this  statement t  involves  the 
property  upon  which  is  based  the  most  convenient  method  of  drawing  a  tangent  to  the  Cartesian,  viz., 
that  the  normal  at  any  point  divides  the  angle  between  the  focal  radii  into  parts  whose  sines  are 
proportional  to  the  factors  of  those  radii  in  the  equation.  If,  then,  we  have  obtained  a  point  G 
on  the  outer  oval  from  the  relation  m  p'  ±  n  p"  -=kc,  we  may  obtain  the  tangent  at  G  by  laying  off 
on  p'  and  p"  distances  proportional  to  m  and  n,  as  Gr  and  G  h,  Fig.  118,  then  bisecting  rh  at  _;' 
and  drawing  the  normal  Gj,  to  which  the  desired  tangent  is  a  perpendicular.- 

At  a  point  on  the  inner  oval  the  distance  would  not  be  laid  off  on.  a  focal  radius  produced,  as 
in  the  case  illustrated. 


.  iso. 


*-" 


'--* 


*  American  Journal  of  Mathematics,   1878. 


f  Salmon.     Higher  Plane   Curves.     Art.    117. 


74 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


CASSIAN     OVALS. 


212.  In  the  Cassian  Ovals  or  Ovals  of  Cassini  the  points  are  connected  with  two  foci  by  the 
relation  p'p"  =  k2,  i.e.,  the  product  of  the  focal  radii  is  equal  to  some  perfect  square.  These  curves 
have  already  been  alluded  to  in  Art.  114  as  plane  sections  of  the  annular  torus,  taken  parallel  to 
its  axis. 


-  121. 


-.  122. 


In  Art.  158  one  form  —  the  Lemniscate  —  receives  special  treatment.  For  it  the  constant  P  must 
equal  w2,  the  square  of  half  the  distance  between  the  foci.  When  k  is  less  than  m,  the  curve 
becomes  two  separate  ovals. 

213.  The  general  construction  depends  on  the  fact  that  in  any  semicircle  the  square  of  an  ordinate 
equals  the  product  of  the  segments  into  which  it  divides  the  diameter.  In  Fig.  122  take  Fl  and 
F2  as  the  foci,  erect  a  perpendicular  F1S  to  the  axis 
Fl  FZ  ,  and  on  it  lay  off  Fl  R  equal  to  the  constant,  k. 
Bisect  FlFt  at  0  and  draw  a  semicircle  of  radius  OR. 
This  cuts  the  axis  at  A  and  B,  the  extreme  points  of 
the  curve;  for  kt  =  FlAxF1B.  Any  other  point  T 
may  be  obtained  by  drawing  from  Fl  a  circular  arc  of 
radius  F^  t  greater  than  Ft  A  ;  draw  t  R,  then  R  x  perpen- 
dicular to  it;  xFl  will  then  be  the  p",  and  Fl  t  the  p',  for 
four  points  of  the  curve,  which  will  be  at  the  intersection  of 
arcs  struck  from  Fl  and  Ft  as  centres  and  with  those  radii. 


To    get    a    normal    at    any   point   T   draw   0  T,   then    make    angle 
the  desired  line. 


^e^F.T  0;    Ts    will    be 


THE     CATENARY. 


214.  If  a  flexible  chain,  cable  or  string,  of  uniform  weight  per  unit  of  length,  be  freely  sus- 
pended by  its  extremities,  the  curve  which  it  takes  under  the  action  of  gravity  is  called  a  Catenary, 
from  catena,  a  chain. 

A  simple  and  practical  method  of  obtaining  a  catenary  on  the  drawing-board,  would  be  to  insert 
two  pins  in  the  board,  in  the  desired  relative  position  of  the  points  of  suspension,  and  then  attach 
to  them  a  string  of  the  desired  length.  By  holding  the  board  vertically,  the  string  would  assume 
the  catenary,  whose  points  could  then  be  located  with  the  pencil  and  joined  in  the  usual  manner 
with  the  irregular  curve.  Otherwise,  if  its  points  are  to  be  located  by  means  of  an  equation,  we 
take  axes  in  the  plane  of  the  curve,  the  y-axis  (Fig.  123)  being  a  vertical  line  through  the  lowest 
point  T  of  "the  catenary,  while  the  x-axis  is  a  horizontal  line  at  a  distance  m  below  T.  The  quan- 
tity m  is  called  the  parameter  of  the  curve,  and  is  equal  to  the  length  of  string  which  represents 
the  tension  at  the  lowest  point. 


THE    CATENARY.— THE    LOGARITHMIC    SPIRAL. 


75 


The  equation   of  the  catenary1  is  then  j/=.^e«  +  e    »)    in    which    e    is    the    base    of   Napierian 

logarithms2   and   has   the   numerical   value   2.7182818 +. 

By  taking  successive   values  of  x  equal  to  m,  2  771,  3  m, 
etc.,  we  get  the  following  values  for  y. 

x=     m...y=-^\e-\ — )  which  for  m  =  unity  becomes  1.54308 

£t      \  &  f 

';   "  "    "     "    3.76217 
"   "  "    "     "   10.0676 

"   "  "    "     "   27.308 

To  construct  the  curve  we  therefore  draw  an  arc  of 
radius  0  B  =  m,  giving  T  on  the  axis  of  y  as  the  lowest 
point  of  the  curve. 

For    x  =05  =  771    we     have     y  =B  P  —  1.54308;     for    x  =•  0  a  =  ^    we   have    y  =  a  n  —  1.03142. 

The   tension   at  any   point  P  is   equal   to   the   weight   of  a   piece   of   rope   of  length  B  P  =  P  C  +  m. 
At  the  lowest   point   the   tangent  is   horizontal.      The   length  of  any  arc   TP  is   proportional  to  the 
angle  6  between   T  C  and   the   tangent  P  V  at  the   upper   extremity   of  the   arc. 

215.  If   a    circle    RLE    be    drawn,    of    radius    equal    to   m,    it    may    be    shown    analytically    that 
tangents  PS    and   QR,   to    catenary   and    circle    respectively,   from    points    at    the    same    level,    will    be 
parallel:    also  that  PS  equals   the   catenary -arc  Pr  T;    S  therefore   traces   the  involute   of  the   catenary, 
and    as  S  B    always    equals  R  0    and    remains    perpendicular    to  PS    (angle   0  R  Q    being   always    90  °) 
we   have   the   curve   TSK  fulfilling  the   conditions  of  a   tractrix.      (See   Art.   202.) 

If  a  parabola,  having  a  focal  distance  m,  roll  on  a  straight  line,  the  focus  will  trace  a  catenary 
having  m  for  its  parameter. 

The  catenary  was  mistaken  by  Galileo  for  a  parabola.  In  1669  Jungius  proved  it  to  be  neither 
a  parabola  nor  hyperbola,  but  it  was  not  till  1691  that  its  exact  mathematical  nature  was  known, 
being  then  established  by  James  Bernouilli. 

THE     LOGARITHMIC     OR     EQUIANGULAR     SPIRAL. 

216.  In  Fig.   124  we  have  the  curve  called  the  Logarithmic  Spiral.     Its  usual  construction  is  based 
on   the   property   that   any   radius   vector,   as   p,   which   bisects  the   angle    between    two   other  radii,   OM 
and  ON,  is  a  mean  proportional  between  them;    i.e.,  p2  =  0  S2  =  0  M  x  0  N. 

If  M  and  G  are  points  of  the  spiral  we  may  find  an  intermediate  point  K  by  drawing  the 
ordinate  OK  to  a  semicircle  of  diameter  OM+OG;  a  perpendicular  through  G  to  G  K  will  then 
give  D,  another  point  of  the  curve,  and  this  construction  may  be  repeated  indefinitely. 

Radii   making   equal   angles   with   each   other   are   evidently   in  geometrical  progression. 

This   spiral  is   often   called   Equiangular  from   the    fact  that    the   angle   is   always   the   same   between 


1  Rankine.    Applied  Mechanics.  Art.  175. 

sin  the  expression  102  =  100  the  quantity  "2"  is  called  the  logarithm  of  100,  It  being  the  exponent  of  the  power  to  which 
10  must  be  raised  to  give  100.  Similarly  2  would  be  the  logarithm  of  64,  were  8  the  base  or  number  to  be  raised  to  the  power 
indicated. 


76 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


a  radius  vector  and  the  tangent    at    its    extremity.      Upon  this  property  is  based  its  use  as  the   out- 
line for  spiral  cams  and  for  lobed  wheels.      The  curve  never  reaches  the  pole. 

The  name  logarithmic  spiral  is  based  on  the  property  that 
the  angle  of  revolution  is  proportional  to  the  logarithm  of  the 
radius  vector.  This  is  expressed  by  p  =  ae,  in  which  6  is  the 
varying  angle,  and  a  is  some  arbitrary  constant. 

To  construct  a  tangent  by  calculation,  divide  the  hyperbolic 
logarithm 1  of  the  ratio  0  M :  0  K  (which  are  any  two  radii 
whose  values  are  known)  by  the  angle  between  these  radii, 
expressed  in  circular  measure;2  the  quotient  will  be  the  tangent 
of  the  constant  angle  of  obliquity  of  the  spiral. 

217.  Among  the  more  interesting  properties  of  this  curve 
are  the  following: 

Its   involute   is   an   equal   logarithmic   spiral. 
Were   a   light   placed   at   the   pole,  the   caustic — whether  by 
reflection    or   refraction  —  would   be   a  logarithmic  spiral. 

The    discovery   of   these    properties    of    recurrence   led   James 
Bernouilli    to    direct  that    this    spiral    be    engraved   on   his   tomb, 
with   the  inscription  —  Eadem  Mutata  Resurgo,  which,  freely  trans- 
lated,  is — /  shall  arise  the  same,  though  changed. 

Kepler  discovered  that  the  orbits  of  the  planets  and  comets  were  conic  sections  having  a  focus 
at  the  centre  of  the  sun.  Newton  proved  that  they  would  have  described  logarithmic  spirals  as 
they  travelled  out  into  space,  had  the  attraction  of  gravitation  been  inversely  as  the  cube  instead  of 
the  square  of  the  distance. 

THE     HYPERBOLIC     OR     RECIPROCAL     SPIRAL. 

218.     In  this   spiral  the  length   of  a   radius   vector   is   in   inverse   ratio  to   the   angle   through  which 
it  turns.      Like   the   logarithmic    spiral,   it   has   an   infinite   number   of 
convolutions   about  the   pole,   which  it  never  reaches. 

The  invention  of  this  curve  is  attributed  to  James  Bernouilli, 
who  showed  that  Newton's  conclusions  as  to  the  logarithmic  spiral 
(see  Art.  217)  would  also  hold  for  the  hyperbolic  spiral,  the  initial 
velocity  of  projection  determining  which  trajectory  was  described. 

To  obtain  points  of  the  curve  divide  a  circle  m58  (Fig.  125) 
into  any  number  of  equal  parts,  and  on  some  initial  radius  Om 
lay  off  some  unit,  as  an  inch ;  on  the  second  radius  0  2  take 

On  On 

c-  ;    on   the   third   — ^-,   etc.     For    one  -  half    the    angle   6    the    radius   vector  would   evidently  be   2  0  n, 
Z  o 

giving   a   point  s  outside   the   circle. 

The  equation  to  the  curve  is  -  =  a  6,  in  which  r  is  the  radius  vector,  a  some  numerical  con- 
stant, and  6  is  the  angular  rotation  of  r  (in  circular  measure)  estimated  from  some  initial  line. 


.  12S- 


'To  get  the  hyperbolic  logarithm  of  a  number  multiply  its  common  logarithm  by  2.3026. 

«ln    circular   measure  360°  =  2irr,  which,  for  r  =  l,  becomes   6.28318;  180  °  =  3.14159 ;   90°  =  1.5708;   60°  =  1.0472;   45°  =0.7854;   30°  = 
0.5236;    1  o  =  0.0174533. 


THE    HYPERBOLIC    SPIRAL.— THE    LITUUS. 


77 


The  curve    has  an    asymptote    parallel    to  the  initial    line,  and    at    a    distance    from    it    equal    to 


—  units, 
a 


To  construct  the  spiral  from  its  equation,  take   0  as  the  pole   (Fig.  26);    0  Q  as  the  initial  line; 
a,  for  convenience,   some  fraction,  as   —  ;   and  as  our  unit  some  quantity,  say  half  an  inch,  that  will 

make  --of  convenient  size.    Then,  taking    Q  0  as  the  initial  line,  make  0  P=  —  =  2",  and  draw  PR 

d  (I 

parallel    to     OQ    for    the    asymptote.      For    6  =  1,    that    is,    for    arc    KH=  radius     OH,    we    have 
r  =  —  =  2",  giving  H  for  one  point  of  the  spiral.     Writing  the  equation  in  the  form   r  =  —  •—  ,    and 

Ut  GJ        0 

expressing  various  values   of  0  in  circular  measure  we  get  the  following  : 

6  =  30°  =  0.5236;  r  =  OM=Z'.'8  +  :        6  =  45°  =  0.7854;  r=0JV=2'/55; 
0  =  90°  =  1.5708;  r  =  OS  =  l'."2+:        6  =180°  =  3.14159;  r=OT=.6366,  etc. 
The   tangent  to  the   curve  at  any  point   makes  with  the  radius  vector   an    angle  <£,  which  is  found 

by  analysis   to    sustain  to  the    angle   6  the  following    trigonometrical    relation,  tun   <£  =  #;   the  circular 

measure  of  0  may  therefore  he  found  in  a  table  of  natural  tangents,  and  the  corresponding  value   of 

<j>  obtained. 

THE    LITUUS.  —  THE    IONIC    VOLUTE. 

\ 

219.     The  Lituus  is  a  spiral  in   which  the  radius  vector  is  inversely  proportional  to  the  square  root 

of  the  angle  through  which  it  has  revolved.    This  relation  is  shown  by  the  equation  r  =  —  -=  ,    also 

a\/  @ 

written  a2  6=  -=- 
r1- 

When  6  =  0  we  find  r  =  oo  ,   which  makes  the  initial  line  an  asymptote  to  the  curve. 

In    Fig.    127    take    0  Q    as    the    initial    line,    0    as    the    pole,  a  =  2,   and    as    our    unit    3"  ;    then 


For  0  =  90°=7r  (in  circular  measure   1.5708)   we  have  r  =  OM=l".  2  +.    For  6  =  1   we    have 
the   radius    0  T  making    an   angle   of  57°.  29  +  with   the   initial   line,   and   in    length    equal   to    -  units, 


78 


THEORETICAL    AND    PRACTICAL     GRAPHICS, 


for    in 


i.  e.,    li".      For    6=45°=-.    (or    0.7854)    r    will    be    OR=1".7+.      Then    0  H  = 

4 

rotating  to   OH  the  radius  vector  passes  over  four  45°  angles,  and  the  radius  must  therefore  be  one- 
half    what    it    was    for    the    first    45°    described. 

Similarly,  QK=  -  — ;    OM  =  --5-,  etc.;   this  rela- 

^  ^j 

tion   enabling   the    student  to    locate  any  number 
of  points. 

To  draw  a  tangent  to  the  curve  we  employ 
the  relation  tan  <f>  =  2  0,  <f>  being  the  angle  made 
by  the  tangent  line  with  the  radius  vector, 
while  6  is  the  angular  rotation  of  the  latter,  in 
circular  measure. 

Architectural  Scrolls. —  The  Ionic  Volute.  The 
Lituus  and  other  spirals  are  occasionally 
employed  as  volutes  and  other  architectural 

ornaments.      In   the   former   application    it   is   customary   for  the   spiral  to   terminate   on   a  circle  called 

the   eye,  into   which   it  blends  tangentially. 

Usually,  in   practice,   circular -arc   approximations   to    true   spiral   forms  are  employed,  the  simplest 

of  which,  for  the   scroll  on  the   capital  of  an  Ionic  column,  is 

probably  the   following: 

Taking  A  0  P,   the   total    height    of   the    volute,    at   sixteen 

of   the    eighteen    "parts"    into    which   the  module   (the   unit   of 

proportion  =  the    semi -diameter    of   the    column)     is     divided, 

draw   the   circular   eye   with   radius  equal  to  one  such  part,  the 

centre    dividing    A  P   into    segments    of    seven    and    nine    parts 

respectively.       Next    inscribe    in    the    eye    a    square    with    one 

diagonal    vertical;     parallel    to    its     sides     draw    (see     enlarged 

square    inn  op)    2 —  4    and    3  —  1,    and    divide    each    into    six 

equal   parts,  which   number   up   to   twelve,   as   indicated.      Then 

(returning    to    main    figure)    the    arc  A  B    has    centre    1    and 

radius    1 — A.      With   2  as   a    centre    draw    arc  BC;    then    CD 

from   centre   3,   etc. 

In  the   complete   drawing   of    an    Ionic    column   the   centre 

of   the    eye    would   be    at    the    intersection    of   a    vertical    line 

from    the    lower    extremity   of   the   cyma  reversa   with    a  hori- 
zontal  through   the   lower   line    of  the    echinus.      To    complete    the    scroll    a    second    spiral    would    be 

required,  constructed   according  to   the   same  law   and   beginning   at   Q,  where  A  Q  is   equal   to   one -half 

part   of  the   module. 


-.  127  (a). 


BRUSH    TINTING    AND    SHADING.  79 


CHAPTER     VI. 

TINTING  —  PLAT     AND     GBADUATED.  —  MASONRY,     TILING,     WOOD     GRAINING,     RIVER-BEDS     AND 
OTHER    SECTIONS,    WITH    BRUSH    ALONE     OR    IN    COMBINED    BRUSH     AND    LINE    WORK. 

220.  Brush-work,  with  ink  or  colors,  is  either  flat  or  graduated.      The  former   gives  the  effect  of 
a    flat   surface   parallel    to   the   paper   on    which   the    drawing   is   made,   while  graded  tints   either  show 
curvature,  or  —  if  indicating  flat  surfaces  —  represent   them  as  inclined   to   the   paper,  i.e.,  to  the  plane 
of  projection.      For  either,  the   paper  should  be,  as  previously  stated  (Arts.  41  and  44)  cold-pressed  and 
stretched. 

The  surface  to  be  tinted  should  not  be  abraded  by  sponge,  knife  or  rubber. 

221.  The    liquid    employed    for    tinting    must    be    free    from    sediment;    or    at    least    the    latter,  if 
present,  must  be  allowed  to  settle,  and  the  brush  dipped  only  in  the  clear  portion  at  the  top.  A  Tints 
may,   therefore,   best  be  mixed  in   an   artist's   water-glass,  rather  than   in   anything  shallower.      In   case 
of  several   colors   mixed   together,   however,  it  would  be  necessary  to  thoroughly  stir  up  the  tint  each 
time  before  taking  a  brushful. 

A  tint  prepared  from  a  cake  of  high-grade  India  ink  is  far  superior  to  any  that  can  be  made 
by  using  the  ready-made  liquid  drawing  inks. 

222.  The    size    of   brush    should    bear    some    relation    to    that    of   the  surface   to   be  tinted;    large 
brushes   for  large   surfaces   and   vice   versa-^*  The  customary  error  of  beginners  is  to   use  too  small  and 
too   dry   a   brush   for  tinting,   and   the   reverse   for  shading. 

223.  Harsh   outlines   are  to   be   avoided   in   brush  work,  especially  in  handsomely  shaded  drawings, 
in  which,   if   sharply   defined,   they   would    detract   from   the   general    effect.      This   will  become   evident 
on   comparing  the   spheres   in   Figs.    1    and   4   of  Plate   II. 

Since  tinting  and  shading  can  be  successfully  done,  after  a  little  practice,  with  only  pencilled 
limits,  there  is  but  little  excuse  for  inking  the  boundaries;  ftytljf,  for  the  sake  of  definiteness,  the 
outlines  are  inked  •**  —  aH-  it  should  be  before  the  tinting,  and  in  the  finest  of  lines,  preferably  of 

Tik;    although  -any—  4nk-  will   do  -  provided  \  soft  sponge   and    plenty   of   clean    water  fee-  Oj" 


applied   to   remove    any   excess    that   will   "run."       The   sponge   is   also   to   be  the   main   reliance  ef---the- 
for  the   correction   of  errors   in    brush    work;    the   water,   however,   and   not  the   friction  to 


be  the   active   agent.       An   entire   tint  may  be  removed  in  this   way   in  case   it   seems   desirable. 

224.     When   beginning   work   incline  the  board    at   a  small   angle,   so   that  the  tint   will   flow   down 
after  the   brush.       For  a  flat,  that  is,   a  uniform  tint,   start  at  the  upper  outline   of  the    surface  to   be 
covered,  and  with  a  brush  full,  yet  not  surcharged  —  which  would  prevent  its  coming  to  a  good  point  — 
pass  lightly   along   from  left  to   right,   and   on   the   return   carry  the   tint   down   a  little   farther,   making 
short,   quick   strokes,   with    the    brush    held    almost    perpendicular    to  the    paper.       Advance  the  tint  as 
evenly    as    possible    along    a    horizontal    line^  wort?    q"rrrck4y-__6fft<;e«i    outlines,    but    more    slowly    along 
outlines,   as   one)  should  ;  never  overrun   th^  latter  and   then    resort    to    "  trimming  "   to    conceal    lack    of 
skill.       It~"ls   possible  for  any   one,   with   care'  and   practice,   to  tint   to  yet  not  over  boundaries. 
-t-    The  advancing  edge  of  the  tint  must  not  be  allowed  to   dry  until  the  lower  boundary  is  reached. 


80 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


No  portion  of  the  paper,  however  small,  should  be  missed  as  the  tint  advances,  as  the  work  is 
lit"1  to  be  spoiled  by  retouching. 

uhould    an$   excess    of   tint    be    found    along   the    lower   edge   of  the  figure  it  should  be   absorbed 
y  the  brush,,  after  first   removing  the  latter's   surplus  by   means   of  blotting  paper. 

-To    get    a    dark    effect    several    medium    tints    laid    on    in    succession,   each    one   drying  before  the 
next  is   applied,   give   better  results   than   one   dark   one. 

The  heightened  effect  described  in  Art.  72,  viz.,  a  line  of  light  on  the  upper  and  left-hand  edges, 
may  be  obtained  either  (a)  by  ruling  a  broad  line  of  tint  with  the  drawing-pen  at  the  desired 
distance  from  the  outline,  and  instantly,  before  it  dries,  tinting  from  it  with  the  brush;  or  (b)  by 
ruling  the  line  with  the  pen  and  thick  Chinese  White. 

225.     A    tint    will    spread    much    more    evenly  on    a   tefge   surface    if  the    paper    be    first  slightly 
dampened  with  clean  water.      As  the  tint  will  follow  the  water,  the  latter  should  be  limited  exactly 
_  to  the  intended  outlines  of  the  final  tint. 


-.  3.3S. 


226.  Of   the    colors    frequently   used    by   engineers    and    architects    those    which   work  best  for  flat 
effects  are  carmine,  Prussian  blue,  burnt  sienna  and   Payne's  gray.     Sepia  and  Gamboge,  are,  fortunately, 
rarely   required    for    uniform    tints;    but    the   former    works    ideally   for    shading   by  the    "dry"   process 
described   in   the  next  article;    and    its    rich    brown    gives    effects    unapproachable    with    anything    else. 
It  has,   however,   this   peculiarity,   that  repeated   touches   upon   a  spot  to    make    it    darker   produce    the 
opposite  effect,  unless  enough  time  elapses  between  the  strokes  to  allow  each  addition  to  dry  thoroughly. 

227.  For  elementary   practice  with   the   brush   the  student  should   lay   flat   washes,   in    India  tints, 
on    from-   six  4e- "ten  -  rectangles,    of    sizes    between  ^J-"  X  $"    and    &^-X  10".       If   successful    with    these 
his    next    work    may    be    the    reproduction    of   Fig.    128,   in    which   H,    V,   P  and    S   denote   horizontal, 
vertical,   profile   and  section  planes  respectively.     The  figure   should   be   considerably   enlarged. 

The   plane  V  may   have   two    washes    of    India    ink;    H   one    of    Prussian    blue;    P    one    of   burnt 
sienna,   and  S  one   of  carmine. 

The   edges   of  the   planes   H,    V  and  P  are   either  vertical   or  inclined  30°   to  the  horizontal. 


BRUSH 


81 


For  the  section -plane   assume  n  and  m  at   pleasure,    giving   direction  nm,  to  which  JR  and 
are    parallel.      A    horizontal,   mz,    through    m    gives    z.       From    w    a    horizontal,    ny,    gives    ?/    <       ab. 
Joining  y  with  z  gives  the  "trace"  of  S  on    V. 

228.     Figures    129    and    130    illustrate    the    use    of   the    brush    in    the    representation    of   masonr, 
The  former  may  be  altogether  in  ink    tints,   or  in  medium    burnt    umber    for    the    front    rectangle    ol 


each  stone,  and  dark  tint  of  the  same,  directly  from  the  cake,  for  the  bevel.  Lightly  pencilled 
limits  of  bevel  and  rectangle  will  be  needed;  no  inked  outlines  required  or  desirable. 

The  last  remark  applies  also  to  Fig.  130,  in  which  "quarry  -faced"  ashlar  masonry  is  represented. 
If  properly  done,  in  either  burnt  umber  or  sepia,  this  gives  a  result  of  great  beauty,  especially 
effective  on  the  piers  of  a  large  drawing  of  a  bridge. 

The  darker  portions  are  tinted  directly  from  the  cake,  and  are  purposely  made  irregular  and 
"jagged"  to  reproduce  as  closely  as  possible  the  fractured  appearance  of  the  stone. 


--  ISO. 


Two  brushes  are  required  when  an  ''over -hang"  or  jutting  portion  is  to  be  represented,  one  with 
a  medium  tint,  the  other  with  the  thick  color,  as  before.  An  irregular  line  being  made  with  the 
latter,  the  tint  is  then  softened  out  on  the  lower  side  with  the  point  of  the  brush  having  the  lighter 
tint.  A  light  wash  of  the  intended  tone  of  the  whole  mass  is  quickly  laid  over  each  stone,  either 
before  or  after  the  irregularities  are  represented,  according  as  an  exceedingly  angular  or  a  somewhat 
softened  and  rounded  effect  is  desired. 


82 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


229.  Designs   in  tiling   are  excellent   exercises,  not  only  for  brush   work   in  flat  tints,  but  also — in 
their   preliminary   construction  —  in   precision   of  line   work.       The   superbly  illustrated   catalogues   of  the 
Minton  Tile  Works   are,  unfortunately,  not  accessible  by  all   students,  illustrating  as   they  do,  the  finest 
and   most   varied   work  in   this  line,  both   of  designer  and   chromo-lithographer;    but   it   is   quite   within 
the   bounds   of  possibility   for  the   careful   draughtsman    to    closely   approach   if    not    equal   the  standard 
and    general    appearance    of   their    work,   and   as   suggestions   therefor   Figs.   131    and   132   are   presented. 

230.  In    Fig.    131    the    upper    boundary,   a  d  h  k,    of   a    rectangle    is    divided    at    a,   b,   c,   etc.,  into 
equal    spaces,   and    through    each   point   of    division    two    lines   are   drawn   with   the   30°   triangle,   as   bx 
and   b  r  through   b.      The   oblique  lines   terminate   on   the   sides   and    lower    line    of   the    rectangle.       If 
the   work  is   accurate — and   it  is   worthless   if  not — any   vertical  line  as  mn,   drawn   through   the   inter- 
section,  m,   of  a  pair  of  oblique   lines,   will   pass  through   the   intersection   of  a   series   of  such   pairs. 


The  figure  shows  three  of  the  possible  designs  whose  construction  is  based  on  the  dotted  lines 
of  the  figure.  For  that  at  the  top  and  right,  in  which  horizontal  rows  of  rhombi  are  left  white,  we 
draw  vertical  lines  as  s  q  and  m  n  from  the  lower  vertex  of  each  intended  white  rhombus,  continuing 
it  over  two  rhombi,  when  another  white  one  will  be  reached.  The  dark  faces  of  the  design  are  to 
be  finally  in  solid  black,  previous  to  which  the  lighter  faces  should  be  tinted  with  some  drab  or 
brown  tint.  The  pencilled  construction  lines  would  necessarily  be  erased  before  the  tint  was  laid  on. 

The  most  opaque  effect  in  colors  is  obtained  by  mixing  a  large  portion  of  Chinese  white 
with  the  water  color,  making  what  is  called  by  artists  a  "body  color."  Such  a  mixture  gives  a 
result  in  marked  contrast  with  the  transparent  effect  of  the  usual  wash;  but  the  amount  of  white 
used  should  be  sufficient  to  make  the  tint  in  reality  a  paste,  and  no  more  should  be  taken  on  the 
brush  at  one  time  than  is  needed  to  cover  one  figure. 

Sepia  and  Chinese  white,  mixed  in  the  proper  proportions,  give  a  tint  which  contrasts  most 
agreeably  with  the  black  and  white  of  the  remainder  of  the  figure.  The  star  design  and  the  hexagons 
in  the  lower  right-hand  corner  result  from  extensions  or  modifications  of  the  construction  just 
described  which  will  become  evident  on  careful  inspection. 


TINTING.  — BRUSH    SHADING. 


83 


231.  Fig.  132  is  a  Minton  design  with  which  many  are  familiar,  and  which  affords  opportunity 
for  considerable  variety  in  finish.  Its  construction  is  almost  self-evident.  (The  equal  spaces,  a  b,  cd, 
mn  —  which  may  be  any  width,  x, —  alternate  with  other  equal  spaces  be,  which  may  preferably  be 
about  3  x  in  width.  Lines  at  45  °,  as  indicated,  complete  the  preliminaries  to  tinting. 


.  132. 


The    octagons    may   be  in    Prussian    blue,   the    hexagons    in    carmine,   and   the   remainder   in   white 
and   black,   as   shown;    or  browns   and   drabs   may   be   employed   for   more  subdued   effects. 


SHADING. 

232.  For    shading,   by   graduated    tints,   provide    a    glass    of    clear    water    m— addition    to   the    tint; 
also   ato'  ample   supply    of  blotting   paper. 

The  water -color  or- ink  tint  may  be  considerably  darker  than  for  flat  tinting;  m-4ac4,  the  darker 
•4t--isy  -provided  it  is  clear,  the  more  rapidly  can  the  desired  effect  be  obtained. 

The   brush   must   contain   much   less   liquid   than   for   flat   work. 

Lay  a  narrow  band  of  tint  quickly  along  the  part  that  is  to  be  the  darkest,  then  dip  the  brush 
into  clear  water  and  immediately  apply  it  to  the  blotter,  both  to  bring  it  to  a  good  point  and  to 
remove  the  surplus  tint.  With— the  j)ow  once -diluted  tint  carry  the  advancing  edge  of  the  band 
slightly  farther.  Repeat  the  operation  until  the  tint  is  no  longer  discernible  as  such. 

The  process  may  be  repeated  from  the  same  starting  point  as  many  times  as  necessary  to 
produce  the  desired  effect;  but  the  work  should  T>e  allowed  to  dry  each  time  before  laying  on  a 
new  tint. 

Any  irregularities  or  streaks  can  easily  be  removed  after  the  work  dries,  by  retouching  or 
"stippling"  with  the  point  of  a  fine  brush  that  contains  but  little  tint — scarcely  more  than  enough 
to  enable  the  brush  to  retain  its  point.  For  small  work,  as  the  shading  of  rivets,  rods,  etc.,  the 
process  just  mentioned,  which  is  also  called  "dry  shading,"  is  especially  adapted,  and,  although 
somewhat  tedious,  gives  the  handsomest  effects  possible  to  the  draughtsman. 

233.  Where   a  good,   general  effect  is   wanted,   to   be   obtained   in  less  time  than  would  be  required 
for  the   preceding   processes,  the  method  of  over-lapping  flat  tints  may  he   adopted.      A  narrower  band 
of  dark   tint   is   first   laid   over  the   part  to   be   the   darkest.      When   dry   this   is   overlaid   by  a  broader 
band   of  lighter  tint.      A   yet   lighter  wash    follows,   beginning   on   the   dark   portion  and   extending  still 
farther  than    its    predecessor.      The    process    is    repeated    with    further    diluted    tints    until    the    desired 
effect  is   obtained. 

Faintly  -  pencilled   lines   may   be   drawn   at  the   outset   as   limits   for  the   edges   of  the  tints. 


84 


THEORETICAL    AND    PRACTICAL    GRAPHIQS. 


This  method  is  better  adapted  for  large  work,  that  is  not  to  be  closely  scrutinized,  than  for 
drawings  that  deserve  a  high  degree  of  finish. 

234.  As  to  the  relative  position  and  gradation  of  the  lights  and  shades  on  a  figure,  the  student 
is  referred  to  Arts.  78  and  79  and  the  chapter  on  shadows;  also  to  the  figures  of  Plate  II,  which 
may  serve  as  examples  to  be  imitated  while  the  learner  is  acquiring  facility  in  the  use  of  the  brush, 
and  before  entering  upon  constructive  work  in  shades  and  shadows.  Fig.  3  of  Plate  II  may  be 
undertaken  first,  and  the  contrast  made  yet  greater  between  the  upper  and  lower  boundaries.  Fig.  1 
(Plate  II)  requires  no  explanation.  In  Fig.  133  we  have  a  wood -cut  of  a  sphere,  with  the  theo- 
retical dark  or  "shade"  line  more  sharply  defined  than  in  the  spheres  on  the  plate. 

-.  133.  ng.  ±3-4. 


'A  drawing  of  the  end  of  a  highly -polished  revolving  shaft,  or  even  of  an  ordinary  metallic  disc, 
would  be  shaded  as  in  Fig.  134. 

Fig.  2  (Plate  II)  represents  the  triangular -threaded  screw,  its  oblique  surfaces  being,  in  mathe- 
matical language,  warped  helicoids,  generated  by  a  moving  straight  line,  one  end  of  which  travels  along 
the  axis  of  a  cylinder  while  the  other  end  traces  or  follows  a  helix  on  the  cylinder. 

The  construction  of  the  helix  having  already  been  given  (Art.  120)  the  outlines  can  readily  be 
drawn.  The  method  of  exactly  locating  the  shadow  and  shade  lines  will  be  found  in  the  chapter 
on  shadows. 

Fig.  4  (Plate  II),  when  compared  with  Fig.  91,  illustrates  the  possibilities  as  to  the 
representation  of  interesting  mathematical  relations.  The  fact  may  again  be  mentioned,  on  the 
principle  of  "line  upon  line,"  as  also  for  the  benefit  of  any  who  may  not  have  read  all  that  has 
preceded,  that  the  spheres  in  the  cone  are  tangent  to  the  oblique  plane  at  the  foci  of  the  elliptical 
section.  The  peculiar  dotted  effect  in  this  figure  is  due  to  the  fact  that  the  original  drawing,  of 
which  this  is  a  photographic  reproduction  by  the  gelatine  process,  was  made  with  a  lithographic 
crayon  upon  a  special  pebbled  paper  much  used  by  lithographers.  The  original  of  Fig.  1,  on  the 
other  hand,  was  a  brush -shaded  sphere  on  Whatman's  paper. 

235.     Fig.   5   (Plate   II)   shows   a   "Phoenix   column,"   the   strongest  form  of  iron  for  a  given  weight, 

for  sustaining  compression.  The  student  is  familiar  with  it  as  an 
element  of  outdoor  construction  in  bridges,  elevated  railroads,  etc.; 
also  in  indoor  work  in  many  of  the  higher  office  buildings  of  our 
great  cities. 

By  drawing  first  an  end  view  of  a  Phoenix  column,  similar  to 
that  of  Fig.  135,  we  can  readily  derive  an  oblique  view  like  that  of 
the  plate,  by  including  it  between  parallels  from  all  points  of  the 
former.  The  proportions  of  the  columns  are  obtainable  from  the 
tables  of  the  company. 

Fig.  135  is  a  cross-section  of  the  8-segment  column,  the  shaded 
portion  showing  the  minimum  and  the  other  lines  the  maximum 
size  for  the  same  inside  diameter. 


^fATERIALS    OF    CONSTRUCTION.  85 


i 


In  a  later  chapter  the  proportions  of  other  forms  of  structural  iron  will  be  found.  Short 
lengths  of  any  of  these,  if  shown  in  oblique  view,  are  good  subjects  for  the  **e-  ass- 

brush,    especially    for    "dry"    shading,    the    effect    to    be    aimed    at    being    that 
of  the  rail   section   of  Fig.    136. 

236.  When  some   particular  material   is   to   be   indicated,  a   flat  tint   of  the 
proper  technical   color   (see   Art.   73)   should   be  laid    on   with   the  brush,   either 
before   or  after  shading.      When   the   latter  is   done    with    sepia    it    is    probably 
safer  to  lay  on  the  flat  tint  first. 

A    darker    tint    of   the    technical   color  should   always   be   given   to   a   cross- 
section.      For  blue -printing,   a  cross -section    may   be    indicated    in    solid    black. 

WOOD. — RIVER  -  BEDS. — MASONRY,    ETC. 

237.  While    the    engineering    draughtsman    is    ordinarily   so    pressed   for  time   as  not  to  be  able  to 
give    his    work    the    highest    finish,    yet    he    ought    to    be    able,    when    occasion    demands,    to     obtain 
both  natural  and  artistic  effects;    and  to  conduce  to  that  end  the  writer  has  taken  pains  to  illustrate 
a    number    of  ways    of  representing   the    materials   of  construction.      Although   nearly  all   of  them   may 
be  —  and   in   the   cuts   are — represented   in  black   and   white   (with   the   exception   of   the    wood -graining 
on   Plate   II),  yet  colors,  in  combined  brush  and  line  work,  are  preferable.      The  student  will,  however, 
need   considerable   practice   with   pen   and   ink  before  it  will  be  worth  while  to  work  on  a  tinted  figure. 

238.  Ordinarily,   in   representing   wood,   the   mere  fact  that   it  is  wood   is    all    that    is    intended    to 
be  indicated.      This  may   be   done   most    simply  by  a   series   of  irregular,   approximately -parallel    lines, 
as  in   Fig.    10   or  as   on  the   rule   in   Fig.  17,  page   12.      Make   no   attempt,  however,  to   have  the  grain 
very  irregular.      The   natural   unsteadiness   of  the   hand,   in    drawing    a    long    line    toward    one    continu- 
ously,  will   cause   almost  all  the  irregularity   desired. 

If  a  better  effect  is  wanted,  yet  without  color,  the  lines  may  be  as  in  Fig.  107,  which  represents 
hard  wood. 

In  graining,  the  draughtsman  should  make  his  lines  toward  himself,  standing,  so  to  speak,  at  the 
end  of  the  plank  upon  which  he  is  working. 

The  splintered  end  of  a  plank  should  be  sharply  toothed,  in  contradistinction  to  a  metal  or 
stone  fracture,  which  is  what  might  be  called  smoothly  irregular. 

239.  An    examination    of   any   piece    of   wood    on    which    the    grain    is    at    all    marked   will   show 
that    it    is    darker    at    the    inner    vertex    of    any    marking    than    at    the    outer    point.       Although    this 
difference    is    more    readily   produced    with    the    brush,   yet    it    may   be  shown   in   a   satisfactory   degree 
with  the   pen,   by   a   series   of  after -touches. 

240.  If  we   fill   the   pen   with   a   rather   dark   tint   of  the   conventional   color,   draw   the  grain  as  in 
the   figures  just   referred  to,   and  then   overlay   all    with  a   medium   flat   wash   of  some  properly   chosen 
color,   we  get   effects   similar  to   those   of  Plate   II. 

On  large  timber -work  the  preliminary  graining,  as  also  the  final  wash,  may  be  done  altogether 
with  the  brush;  as  was  the  original  of  Fig.  9,  Plate  II. 

End  views  of  timbers  and  planks  are  conventionally  represented  by  a  series  of  concentric  free- 
hand rings  in  which  the  spacing  increases  with  the  distance  from  the  heart;  these  are  overlaid  with 
a  few  radial  strokes  of  darker  tint.  In  ink  aione  the  appearance  is  shown  in  Figs.  39  and  115. 

241.  The   color -mixtures   recommended   by  different  writers  on  wood  graining  are   something  short 
of   infinite    in    number;    but   with    the   addition   of  one   or  two   colors  to   those  listed  in   the   draughts- 
man's outfit   (Art.  56)   one  should  be  able  to  imitate  nature's  tints  very  closely. 


86 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


.  137-. 


No  hard-and-fast  rule  as  to  the  proportions  of  the  colors  can  be  given.  In  this  connection  we 
may  quote  Sir  Joshua  Reynolds'  reply  to  the  one  who  inquired  how  he  mixed  his  paints.  "With 
brains,"  said  he.  One  general  rule,  however;  always  employ  delicate  rather  than  glaring  tints. 

Merely  to  indicate  wood  with  a  color  and  no  graining  use  burnt  sienna,  the  tint  of  Figs  7,  8 
and  10  of  Plate  II. 

Drawing  from  the  writer's  experience  and  from  the  suggestions  of  various  experimenters  in  this 
line  the  following  hints  are  presented:  — 

In  every  case  grain  first,  then  overlay  with  the  ground  tint,  which  should  always  be  much  lighter 
than  the  color  used  for  the  grain.  If  possible  have  at  hand  a  good  specimen  of  the  wood  to  be 
imitated. 

Hard  Pine:  Grain — burnt  umber  with  either  carmine  or  crimson  lake;  for  overlay  add  a  little 
gamboge  to  the  grain -tint  diluted. 

Soft  Pine:     Gamboge   or  yellow   ochre   with   a   small   amount   of  burnt  sienna. 

Black  Walnut:  Grain — burnt  umber  and  a  very  little  dragon's  blood;  final  overlay  of  modified 
tint  of  the  same  or  with  the  addition  of  Payne's  gray. 

Oak:  Grain — burnt  sienna;  for  overlay,  the  same,  with 
yellow  ochre. 

Chestnut:  Grain — burnt  umber  and  dragon's  blood;  over- 
lay of  the  same,  diluted,  and  with  a  large  proportion  of  gam- 
boge or  light  yellow  added. 

Spruce:  Grain  —  burnt  umber,  medium;  add  yellow  ochre 
for  the  overlay. 

Mahogany:  Grain — burnt  sienna  or  umber  with  a  small 
amount  of  dragon's  blood;  dilute,  and  add  light  yellow  for 
the  overlay. 

Rosewood:  Grain — replace  the  dragon's  blood  of  mahogany- 
grain  by  carmine,  and  for  overlay  dilute  and  add  a  little 
Prussian  blue. 

242.  River-beds    in    black    and    white    or   in     colors    have 
been   already   treated   in    Art.   26,   to    which    it    is    only   neces- 
sary to  add  that   such  sections  are  usually  made  quite  narrow, 
and,    preferably — if   in    color — shaded    quite    abruptly    on    the 
side   opposite  the   water. 

243.  The   sections   of  masonry,  concrete,  brick,  glass  and  vul- 
canite, given   on   page   25   as   pen   and   ink    exercises,   are  again 

presented  in   Fig.    137,   for  reproduction   in    combined    brush    and    line 
work.      The   appropriate   color  is   indicated   under   each   section. 

244.     Masonry    constructions    may   be    broadly    divided    into    rubble 
and   ashlar. 

In  ashlar  masonry  the  bed -surfaces  and  the  joints  (edges)  are 
shaped  and  dressed  with  great  care,  so  that  the  stones  may  not 
only  be  placed  in  regular  layers  or  courses,  but  often  fill  exactly 
some  predetermined  place,  as  in  arch  construction,  in  which  case  the  determination  of  their  forms 
and  the  derivation  of  the  patterns  for  the  stone-cutter  involves  the  application  of  the  Descriptive 
Geometry  of  Monge.  (Art.  283). 


REPRESENTATION    OF    MASONRY. 


87 


Fig.   1-40. 


Rubble  work,  however,  consists  of  constructions  involving  stones  mainly  "in  the  rough,"  but  may 
be  either  coursed  or  uncoursed.  Fig.  138  is  a  neat  example  of  uncoursed  though  partially  dressed 
or  "hammered"  rubble.  In  section,  as  shown  in  Fig.  137,  it  is  merely  necessary  to  rule  section- 
lines  over  the  boundaries  of  the  stones  —  a  remark  applying  equally  to  ashlar  masonry. 

The  other  examples  in  this  chapter 

are    of    ashlar,    mainly    "  quarry  -  faced," 

that   is,  with   the   front  nearly  as  rough 

as      when      quarried.       A     beveled      or 

"chamfered"   ashlar   is    shown    in    Figs. 

129   and  140,  the  latter  shaded  in   what 

is   probably   the  most   effective   way   for 

small    work,   viz.,    with    dots,    the    effect 

depending    upon    the    number,    not    the 

size   of  the   latter. 

Only   a   careful    examination   of  the 
kind  and  position  of  the   lines  in  the   other  figures   on   this   page  will  disclose   the   secret  of  the  variety 
in    the    effects    produced.       For    the    handsomest    results    with    any    of   these    figures    the    pen-work — 


whether    dotting    or    "cross-hatching"  —  should    be    preceded    by    an    undertone    of    either    India    ink, 
umber,    Payne's    gray,    cobalt   or    Prussian    blue,    according    to    the    kind    of    stone    to    be    represented. 


.  143. 


For    slate    use    a    pale    blue;    for    brown    free -stone    either    an    umber    or    sepia;    while    for    stone    in 
general,   kind  immaterial,   use  India  ink. 


•88  THEORETICAL    AND    PRACTICAL    GRAPHICS. 


CHAPTER     VII. 

FREE-HAND    AND    MECHANICAL     LETTERING.  — PROPORTIONING    OF    TITLES. 

245.  Practice    in    lettering    forms    an    essential    part    of    the    elementary   work    of    a    draughtsman. 
Every   drawing  has   to   have  its  title,   and  the   general   effect   of   the   result  as   a   whole   depends   largely 
upon  the   quality   of  the   lettering. 

Other  things  being  equal,  the  expert  and  rapid  draughtsman  in  this  line  has  a  great  advantage 
over  one  who  can  do  it  but  slowly.  For  this  reason  free-hand  lettering  is  at  a  high  premium,  and 
the  beginner  should,  therefore,  aim  not  only  to  have  his  letters  correctly  formed  and  properly  spaced, 
but,  as  far  as  possible,  to  do  without  mechanical  aids  in  their  construction.  When  under  great 
pressure  as  to  time  it  is,  however,  perfectly  legitimate  to  employ  some  of  the  mechanical  expedients 
used  in  large  establishments  as  "short  cuts"  and  labor-savers.  Among  these  the  principal  are 
"  tracing "  and  the  use  of  rubber  types. 

246.  To  trace  a  title  one  must  have  at  hand  complete  printed  alphabets  of  the  size  of  type  required. 
Placing  a   piece    of   tracing-paper    over    the    letter    wanted,   it  is   traced   with   a  hard  pencil,   the  paper 
then  slipped  along  to  the  next  letter  needed,  and  the  process  repeated  until   the  words  desired    have 
been    outlined.      The    title   is  then   transferred   to   the   drawing    by    first    running   over  the   lines  on   the 
back   of   the  tracing-paper  with   a  soft   pencil,   after  which   it  is   only   necessary   to   re-trace   the   letters 
with   a   hard   pencil,   on  the  face    of   the    transfer -paper,   to    find    their    outlines    faintly   yet   sufficiently 
indicated   on   the   paper   underneath.      Carbon   paper   may  also   be   used   for  transferring. 

247.  The    process   just    described    would    be   of   little    service    to   a   ready   free-hand    draughtsman, 
but    with    the    use    of   rubber  types,  for    the    words    most    frequently    recurring    in    the    titles,   a    merely 
average    worker    may    easily    get   results   which — in   point    of    time  —  cannot   be   exceeded   by   any   other 
method.      When    employing    such    types    either    of   the    following  ways   may   be   adopted :     (a)    a    light 
impression   may    be    made    with  the    aniline   ink   ordinarily    used    on    the    pads,   and   the   outlines   then 
followed   and   the   "filling   in"   done   either   with   a    writing- pen*   or   fine- pointed    sable -hair    brush;     or 
(b)  the  impression   may   be   made   after  moistening  the  types   on   a   pad   that  has   been  thoroughly    wet 
with    a    light    tint    of  "India    ink.      The    drawing-ink    must    then    be    immediately    applied,    free-hand, 
with   a   Falcon   pen   or  sable  brush,  before  the   type -impression   can   dry.     The   pen   need  only  be  passed 
down   the   middle   of  a  line,   as  on  the  dampened  surface  the   ink  will  spread  instantly  to   the  outlines. 

248.  The   educated   draughtsman   should,  however,   be  able   not   only   to   draw   a   legible  title  of  the 
simple    character    required    for    shop -work,   and    in    which    the    foregoing    expedients   would    be   mainly 
serviceable,   but  be  prepared  also  for  work   out   of  the   ordinary   line,   and,   if    need   be,   quite   elaborate, 
as    on    a    competitive    drawing.      Such    knowledge    can    only    be  gained    by  careful   observation    of   the 
forms   of  letters,   and   considerable   practice  in  their   construction. 

No    rigid    rules  can    be  laid    down    as    to    choice    of    alphabets    for    the    various    possible    cases. 

Common -sense,   custom   and  a  natural   regard   for  the   "fitness    of   things"   are   the    determining   factors. 

Obviously   rustic  letters   would   be   out   of  place   on   a    geometrical    drawing,   and  other  incongruities 


1  Ket'er  to  Art.  27  with  regard  to  the  pens  to  be  used  for  the  various  styles  of  letters. 


DESIGNING     OF    TITLES.  89- 

will  naturally  suggest  themselves.  In  addition  to  the  hints  in  Art.  27  a  few  general  principles  and 
methods  may,  however,  be  stated  to  the  advantage  of  the  beginner,  who  should  also  refer  to  the 
special  instructions  given  in  connection  with  certain  specimen  alphabets  at  the  end  of  this  work. 

249.  In  the  first  place,  a  title  should  be  symmetrical  with  respect  to  a  vertical  centre-line,  a 
rule  which  should  be  violated  but  rarely,  and  then,  usually,  when  the  title  is  to  be  somewhat  fancy 
in  design,  as  for  a  magazine  cover. 


Plates 

BOlAIIIOAt  DRAWING 

drawn  by  Qlorflatttif  Ban  dtofear  ^  ^ 

LEADING     TECHNICAL     SCHOOL 

Jan.  —  June,    3OO1. 

250.  If  it  be  a  complete  as  distinguished  from  a  partial  or  sub-  title  it  will  answer  the  following 
questions  which  would  naturally  arise  in  the  mind  of  the  examiner:  — 

What  is   it?  —  Where   done?  —  By   whom  ?  —  When  ?  —  On   what  scale? 

In  answering  these  questions  the  relative  valuation  and  importance  of  the  lines  are  expressed  by 
the  sizes  and  kinds  of  type  chosen.  This  is  a  point  requiring  most  careful  consideration,  as  the  final 
effect  depends  largely  upon  a  proper  balancing  of  values. 


OF 


PERFECTION  SUSPENSION   BRIDGE 

—  «>*«  da  signed, 


Dnndwin,   Mackenzie   %>   Cartwright 

-  •>»  <    -e  MINNEAPOLIS,    MINN.  •  •    »  K. 


SCALE    4   FT.  =    I    IN.       JtinS      1©,       2©OO.       JOSE    MARTINEZ.    DEL. 

251.  The  "By  whom?"  may  cover  two  possibilities.  In  the  case  of  a  set  of  drawings  made  in 
a  scientific  school  it  would  refer  to  the  draughtsman,  and  his  name  might  properly  have  considerably 
greater  prominence  than  in  any  other  case.  The  upper  title  on  this  page  is  illustrative  of  this  point, 
as  also  of  a  symmetrical  and  balanced  arrangement,  although  cramped  as  to  space,  vertically. 

Ordinarily  the  "  By  whom  ?  "  will  refer  to  the  designer,  and  the  draughtsman's  name  ought  to 
be  comparatively  inconspicuous,  while  the  name  of  the  designer  should  be  given  a  fair  degree  of 
prominence.  This,  and  other  important  points  to  be  mentioned,  are  illustrated  in  the  preceding 


90  THEORETICAL    AND    PRACTICAL     GRAPHICS. 

arrangement,   printed,   like   the   upper  title,   from   types    of   which   complete   alphabets   will    be   found   at 
the   end   of  this   work. 

252.  The  abbreviation  Del.,  often   placed   after  the   draughtsman's   name,  is   for   Delineavit  —  He  drew 
it  —  and    does    not    indicate    what    the    visitor    at    the   exhibition   supposed,   that  all    good    draughtsmen 
hail    from   Delaware. 

253.  The  best  designed   titles   are   either  in   the   form    of   two    truncated   pyramids   having,    if    pos- 
sible,  the   most  important  line  as   their  common  base,   or   else   elliptical  in   shape. 

254.  The  use   of  capitals   throughout    a    line   depends   upon   the  style    of   type.      It    gives    a    most 
unsatisfactory   result  if  the  letters   are   of  irregular   outline,   as   is  amply   evidenced  by   the   words 


each   letter   of  which   is   exquisite  in   form,   but   the   combination   almost    illegible.      Contrast   them   with 
the   same   style,   but  in   capitals   and   small  letters:  — 


Utefjatttcal 


255.  As  to  spacing,  the  visible  white  spaces  between  the  letters  should  be  as  nearly  the  same 
as  possible.  In  this  feature,  as  in  others,  the  draughtsman  can  get  much  more  pleasing  results  than 
the  printer,  since  the  latter  usually  has  each  letter  on  a  separate  piece  of  metal,  and  can  not  adjust 
his  space  to  any  particular  combination  of  letters,  such  as  FA,  L  V,  W  A  or  A  V,  where  a  better 
effect  would  be  obtained  by  placing  the  lower  part  of  one  letter  under 
the  upper  part  of  the  next.  This  is  illustrated  in  Fig.  146,  which  may  "v'V" 
be  contrasted  with  the  printer's  best  spacing  of  the  separate  types  for  /  /  \  *  /  H 

t,hf>    A     and    W    in    thfi    word     "  Drawings  "    nf   thfi    last,    titlfi  -/    -/   — -I — — J      J 


the  A   and   W  in  the  word   "  Drawings  "   of  the   last   title. 

256.  The  amount  of  space  between  letters  will  depend  upon  the  length  of  line  that  the  word  or 
words  must  make.  If  an  important  word  has  few  letters  they  should  be  "  spaced  out,"  and  the 
letters  themselves  of  the  "  extended  "  kind,  i.  e.,  broader  than  their  height.  The  following  word  will 
illustrate.  The  characteristic  feature  of  this  type,  viz.,  heavy  horizontals  and  light  verticals,  is  com- 
mon to  all  the  variations  of  a  fundamental  form  frequently  called  Italian  Print. 


When,  on  the  other  hand,  many  letters  must  be  crowded  into  a  small  space,  a  "  condensed  " 
style  of  letter  must  be  adopted,  of  which  the  following  is  an  example: 

Pennsylvania  Railroad. 

257.  While  the  varieties  of  letters  are  very  numerous  yet  they  are  all  but  changes  rung  on  a 
few  fundamental  or  basal  forms,  the  most  elementary  of  which  is  the 

GOTHIC,  ALSO  CALLED  HALF-  BLOCK. 

Letters  like  B,  0,  etc.,  which  have,  usually,  either  few  straight  parts  or  none  at  all,  may,  for 
the  sake  of  variety  as  also  for  convenience  of  construction,  be  made  partially  or  wholly  angular;  in 
the  latter  case  the  form  is  called  Geometric  Gothic  by  some  type  manufacturers.  It  is  only  appropriate 
for  work  exclusively  mechanical.  The  rounded  forms  are  preferable  for  free-hand  lettering. 


LETTERING. 


91 


The    following    complete    Gothic    alphabet    is    so    constructed    that    whether    designed    in   its 
densed"   or   "extended"  form   the   proper  proportions   may   be   easily   preserved. 


con- 


* IM 


f  V  1  \  U  i. 


Taking  all  the  solid  parts  of  the  letters  at  the  same  width  as  the  I,  we  will  find  any  letter  of 
average  width,  as  U,  to  be  twice  that  unit,  plus  the  opening  between  the  uprights,  which  last,  being 
indeterminate,  we  may  call  x,  making  it  small  for  a  "condensed"  letter,  and  broad  as  need  be  for 
an  "extended"  form. 

The  word  march  would  foot  up  5  U  +  3,  disregarding- — as  we  would  invariably — the  amount  the 
foot  of  the  R  projects  beyond  the  main  right-hand  outline  of  the  letter.  In  terms  of  x  this  makes 
5  x  +  13,  as  U  =  x  +  2.  Allowing  spaces  of  1|  unit  width  between  letters  adds  5  to  the  above,  making 
5  x  4- 18  for  the  total  length  in  terms  of  the  I.  Assuming  x  equal  to  twice  the  unit  we  would 
have  the  whole  word  equal  to  twenty -eight  units;  and  if  it  were  to  extend  seven  inches  the  width 
of  the  solid  parts  would  therefore  be  one -quarter  of  an  inch. 

Where  the  width  of  a  letter  is  not  indicated  it  is  assumed  to  be  that  of  the  U.  The  W  is 
equal  to  2U —  1.  This  relation,  however,  does  not  hold  good  in  all  alphabets. 

The   angular   corners   are   drawn   usually   with   the   45°   triangle. 

The  guide-lines  show  what  points  of  the  various  letters  are  to  be  found  on  the  same  level,  and 
should  be  but  faintly  pencilled. 

As  remarked  in  Art.  27,  the  extended  form  of  Gothic  is  one  of  the  best  for  dimensioning  and 
lettering  working  drawings,  and  is  rapidly  coming  into  use  by  the  profession. 

258.  The  Full -Block  letter  next  illustrated  is  easier  to  work  with  than  the  Gothic  in  the  matter  of 
preliminary   estimate,   as   the   width    of  each    letter — in   terms   of   unit  squares — is   evident  at  a  glance. 

The  same  word  march  would  foot  up  twenty -seven  squares  without  allowing  for  spaces  between 
letters.  Calling  the  latter  each  two  we  would  have  thirty -five  squares  for  the  same  length  as  before 
(seven  inches),  making  one -fifth  of  an  inch  for  the  width  of  the  solid  parts.  For  convenience  the 
widths  of  the  various  letters  are  summarized: 

1  =  3;   C,G,O,Q,S,Z  =  4;   A,  B,  D,E,F,  J,  L,  P,  R,T,&  =  5;   H,K,N,U,  V,X,  Y  =  6;  M  =  7;   W  =  8. 

259.  In   case   the    preliminary   figuring    were    only  approximate    and    there   were  but  two   words  in 
the    line,   as,   for    example,  Mechanical  Drawing,   a    safe    method    of    working    would    be   to   make   a   fair 
allowance   for  the   space  between  the   words,   begin    the    first    word    at    the    calculated    distance    to    the 
left   of  the   vertical   centre-line,   complete  it,   then   work  the  second   word   backward,  beginning  with  the 


92 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


G  as  far  to  the  right  of  the  reference  line  as  the  M  was  to  the  left.  On  completing  the  second 
word  any  difference  between  the  actual  and  the  estimated  length  of  the  words,  due  to  over-  or 
under- width  of  such  letters  as  M,  W  and  I,  will  be  merged  into  the  space  between  the  words. 


H-Y-KrvI- vi^rsH-i-i-:-- -  •'•-<•' •!- 
fT+^H'-rrtH-F'-iv-F--'--^ 


With  three  words  in  a  line  the  same  method  might  be  adopted,  the  middle  word  being  easily 
placed  half  way  between  the  others,  which,  by  this  method  of  construction  would  not  only  begin 
correctly  but  also  terminate  where  they  should. 

260.  Note  particularly  that  the  top   of  a   B  is  always  slightly  smaller   than   the  bottom; 
similarly  with  the  S.      This  is  made  necessary  by  the   fact  that  the   eye  seems   to   exaggerate 
the   upper  half  of  a   letter.      To  get   an  idea   of  the  amount  of  difference   allowable   compare 
the  foUowing  equal  letters  printed  from   Roman   type,  condensed.     Although  not  so  important 

in  the   E,   some   difference   between  top   and   bottom   may   still   to   advantage  be  made.      Another   refine- 
ment is   the   location   of  the  horizontal   cross-bar   of  an   A   slightly   below  the  middle   of  the  letter. 

261.  While  vertical    letters   are    most    frequently  used,  yet    no    handsomer    effect    can    be    obtained 
than   by   a   well  -  executed   inclined   letter.      The  angle   of  inclination  should   be  about  70°. 

Beginners    usually    fail    sadly    in    their    first    attempt 

with   the   A  and    V,   one    of    whose    sides    they    give    the 

same  slant   as  the  upright   of  the  other  letters.      In  point 

of    fact,    however,    it    is    the    imaginary    (though,    in    the 

construction,  pencilled)  centre-line  which   should  have  that 

inclination.      See   Fig.   150. 

In    these    forms — the    Roman    and    Italic    Roman — the   union   of  the   light   horizontals   or   "seriffs" 

with   the   other  parts   is   in  general   effected    by  means    of   fine    arcs,   called   "fillets,"   drawn    free-hand. 

On  many  letters   of   this    alphabet    some    lines    will,    however,    meet    at    an    angle,    and    only  a    careful 

examination   of   good   models   will   enable   one   to   construct   correct  forms.      Upon   the   size  of  the  fillets 

the  appearance  of  the  letter  mainly  depends,  as  will  be  seen  by  a  glance  at  Fig.  151,  which  repro- 
duces, exactly,  the  N  of  each  of  two  leading  alphabet  books.  If  the  fillets 
round  out  to  the  end  of  the  spur  of  the  letter,  a  coarse  and  bulky  appear- 
ance is  evidently  the  result;  while  a  fine  curve,  leaving  the  straight 
horizontals  projecting  beyond  them,  gives  the  finish  desired.  This  is  further 

illustrated  by   No.   23  of  the  alphabets  appended,  a  type  which  for  clearness  and  elegance  is  a  triumph 

of  the  founder's  art.     As  usually  constructed,  however,  the  D  and  R  are  finished  at  the  top  like  the  P. 


-.  isi. 


i3Hl^. 
fe} 


94  THEORETICAL    AND    PRACTICAL     GRAPHICS. 

262.  The   Roman  alphabet    and    its    inclined    or    italic  form   are  much   used   in   topographical  work. 
A    text -book    devoted    entirely  to   the    Roman  alphabet   is   in  the  market,   and  in   some   works   on 

topographical  drawing  very  elaborate  tables  of  proportions  for  the  letters  are  presented;  these  answer 
admirably  for  the  construction  of  a  standard  alphabet,  but  in  practice  the  proportions  of  the  model 
would  be  preserved  by  the  draughtsman  no  more  closely  than  his  ?ye  could  secure.  Usually  the 
small  letters  should  be  about  three- fifths  the  height  of  the  capitals.  Except  when  more  than  one- 
third  of  an  inch  in  height  these  letters  should  be  entirely  free-hand. 

263.  When  a  line  of  a  title  is  curved  no   change   is   made   in   the  forms    of   the    letters;    but    if    of 
a    vertical,  as    distinguished    from    a    slanting    or    italic    type,   the    centre-line    of   each   letter    should,   if 
produced,   pass   through   the   centre   of  the   curve. 

Italic  letters,  when  arranged  on  a  curve,  should  have  their  centre-lines  inclined  at  the  same  angle 
to  the  normal  (or  radius)  of  the  curve  as  they  ordinarily  make  with  the  vertical. 

264.  An  alphabet  which  gives   a  most   satisfactory   appearance,   yet    can    be    constructed   with   great 
rapidity,   is   what  we   may  call    the    "Railroad"    type,   since    the    public    has    become   familiar    with    it 
mainly   from   its   frequent   use  in   railroad   advertisements. 

The  fundamental  forms  of  the  small  letters,  with  the  essential  construction  lines,  are  given  in 
rectangular  outline  in  the  complete  alphabet  on  the  preceding  page,  with  various  modifications  thereof 
in  the  words  below  them,  showing  a  large  number  of  possible  effects. 

At  least  one  plain  and  fancy  capital  of  each  letter  is  also  to  be  found  on  the  same  page,  with  in 
some  instances  a  still  larger  range  of  choice. 

No  handsomer  effects  are  obtainable  than  with  this  alphabet,  when  brush  tints  are  employed  for 
the  undertone  and  shadows. 

265.  For  rapid    lettering    on   tracing -cloth,   Bristol 
board  or  any  smooth -surfaced  paper  a  style  long  used 
abroad  and  increasing  in  favor  in  this  country  is  that 
known   as   Round    Writing,  illustrated  by  Fig.  152,  and 

for    which    a    special    text -book    and    pens    have    been    prepared    by    F.    Soennecken.       The    pens    are 

stubs    of   various    widths,   cut    off    obliquely,   and    when    in 
use   should   not,  as   ordinarily,  be   dipped   into   the   ink,  but 
the    latter    should    be    inserted,   by   means    of   another    pen, 
between    the    top    of    the    Soennecken    pen    and    the    brass 
"feeder"  that  is  usually  slipped  over  it  to  regulate  the  flow. 
The    Soennecken    Round    Writing   Pens   are   also   by   far 
the    best    for    lettering    in    Old    English,    German    Text    and 
kindred  types. 
The    improvement    due    to    the    addition    of   a  few   straight  lines  to   an   ordinary   title  will   become 

evident  by   comparing   Figs.  ' 

153  and  154.     The  judicious 

use    of    "word    ornaments," 

such   as   those    of   alphabets 

33,  42,  49,  and  of  several  of 

the    other    forms   illustrated, 


,h< 


eehaniea 


•Meerran  l 


will     greatly     enhance     the  |  &  C  I"]  Q  H  I  Q  Q 

appearance   of  a  title    with- 
out materially  increasing  the   time   expended   on   it.      This   is   illustrated   in  the  lower  title  on  page  89. 


DESIGNS    FOR    BORDERS. 


95 


96 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


266.  Borders.      Another  effective  adjunct  to  a  map  or  other  drawing  is  a  neat  border.     It  should 
be  strictly  in  keeping  with   the   drawing,   both   as   to   character  and   simplicity. 

On  page  95  a  large  number  of  corner  designs  and  borders  is  presented,  one-third  of  them  orig- 
inal designs,  by  the  writer,  for  this  work.  The  principle  of  their  construction  is  illustrated  by  Fig. 
155,  in  which  the  larger  design  shows  the  necessary  preliminary  lines,  and  the  smaller  the  complete 
corner.  It  is  evident  in  this,  as  in  all  cases  of  interlaced  designs,  that  we  must  first  lay  off  each 
way  from  the  corner  as  many  equal  distances  as  there  are  bands  and  spaces,  and  lightly  make  a 
network  of  squares  —  or  of  rhombi,  if  the  angles  are  acute  —  by  pencilled  construction  -lines  through 
the  points  of  division. 

267.  Shade    lines   on    borders.      The   usual    rule    as    to    shade  lines   applies   equally  to  these   designs, 
thus:      Following    any    band    or    pair    of    lines    making    the    turns    as    one 

piece,  if  it  runs  horizontally  the  lower  line  is  the  heavier,  while  in  a 
vertical  pair  the  right-hand  line  is  the  shaded  line.  This  is  on  the 
assumption  that  the  light  is  coming  in  the  direction  usually  assumed  for 
mechanical  drawings,  i.  e.,  descending  diagonally  from  left  to  right. 

In  case  a  pair  of  lines  runs  obliquely,  the  shaded  lines  may  be 
determined  by  a  study  of  their  location  on  the  designs  of  the  plate  of 
borders. 

It  need  hardly  be  said  that  on  any  drawing  and  its  title  the  light 
should  be  supposed  to  come  from  but  one  'direction  throughout,  and  not  be 
shifted;  and  the  shaded  lines  should  be  located  accordingly.  This  rule  is  always  imperative. 

In  drawing  for  scientific  illustration  or  in  art  work  it  is  allowable  to  depart  from  the  usual 
strictly  conventional  direction  of  light,  if  a  better  effect  can  thereby  be  secured. 

268.  A  striking  letter  can  be  made  by  drawing  the  shade  line  only,  as  in  Fig.  146,  page  90,  which 
we    may  call  "Full  -Block    Shade  -Line,"  being    based    upon  the    alphabet  of   Fig.   148,  page  92,  as  to 
construction.      Owing  to  its  having  more  projecting  parts  it  gives  a  much    handsomer   effect  than  the 


The  student  will  notice  that  the  light  comes  from  different  directions  in  the  two  examples. 

These  forms  are  to  the  ordinary  fully  -outlined  letters  what  art  work  of  the  "impressionist" 
school  is  to  the  extremely  detailed  and  painstaking  work  of  many;  what  is  actually  seen  suggests 
an  equal  amount  not  on  the  paper  or  canvas. 

269.  While  a  teacher  of  draughting  may  well  have  on  hand,  as  reference  works  for  his  class, 
such  books  on  lettering  as  Prang's,  Becker's  and  others  equally  elaborate,  yet  they  will  be  found  of 
only  occasional  service,  their  designs  being  as  a  rule  more  highly  ornate  than  any  but  the  specialist 
would  dare  undertake,  and  mainly  of  a  character  unsuitable  for  the  usual  work  of  the  engineering  or 
architectural  draughtsman,  whose  needs  were  especially  in  mind  when  selecting  types  for  this  work. 

The  alphabets  appended  afford  a  large  range  of  choice  among  the  handsomest  forms  recently 
designed  by  the  leading  type  manufacturers,  also  containing  the  best  among  former  types;  and  with 
the  "Railroad,"  Full  -Block  and  Half-  Block  alphabets  of  this  chapter,  proportioned  and  drawn  by  the 
writer,  supply  the  student  with  a  practical  "stock  in  trade"  that  it  is  believed  will  require  but 
little,  if  any,  supplementing. 


COPYING    PROCESSES.  — DRAWING    FOR    ILLUSTRATION.  97 


CHAPTER     VIII. 

BLUE -FEINT    AND    OTHER    COPYING    PROCESSES.— METHODS    OF    ILLUSTRATION. 

270.  While    in    a    draughting    office    the    process   described    below   is,   at   present,   the   only   method 
of  copying   drawings    with    which    it   is   absolutely  essential   that    the   draughtsman    should    be   thoroughly 
acquainted,   he    may,   nevertheless,    find    it    to    his    advantage    to    know    how    to    prepare    drawings    for 
reproduction    by   some   of   the    other    methods    in    most    general    use.      He    ought    also    to    be    able    to 
recognize,   usually,  by  a  glance  at   an  illustration,  the   method   by   which   it   was   obtained.     Some   brief 
hints   on  these   points   are   therefore  introduced. 

Obviously,  however,  this  is  not  the  place  to  give  full  particulars  as  to  all  these  processes,  even 
were  the  methods  of  manipulation  not,  in  some  cases,  still  "trade  secrets";  but  the  important  details 
concerning  them,  that  have  become  common  property,  may  be  obtained  from  the  following  valuable 
works:  Modern  Heliographic  Processes*  by  Ernst  Lietze;  Photo -Engraving,  Etching  and  Lithography  ,t  by 
W.  T.  Wilkinson;  Modern  Reproductive  Graphic  Processes,"  by  Jas.  S.  Pettit,  and  Photo -Engraving,  by 
Carl  Schraubstadter,  Jr. 

THE     BLUE -PRINT     PROCESS. 

271.  By  means   of  this   process,  invented   by  Sir  John  Herschel,  any  number  of  copies  of  a  draw- 
ing  can   be   made,   in   white   lines   on   a   blue   ground.      In   Arts.   43   and   45   some   hints   will   be   found 
as   to   the   relative   merits   of  tracing -cloth   and   "Bond"   paper,   for  the   original   drawing. 

A  sheet  of  paper  may  be  sensitized  to  the  action  of  light  by  coating  its  surface  with  a  solution 
of  red  prussiate  of  potash  (ferri  cyanide  of  potassium)  and  a  ferric  salt.  The  chemical  action  of 
light  upon  this  is  the  production  of  a  ferrous  salt  from  the  ferric  compound;  this  combines  with 
the  ferricyanide  to  produce  the  final  blue  undertone  of  the  sheet;  while  the  portions  of  the  paper 
from  which  the  light  was  intercepted  by  the  inked  lines,  become  white  after  immersion  in  water. 

The  proportions  in  which  the  chemicals  are  to  be  mixed  are,  apparently,  a  matter  of  indiffer- 
ence, so  great  is  the  disparity  between  the  recipes  of  different  writers;  indeed,  one  successful 
draughtsman  says:  "Almost  any  proportion  of  chemicals  will  make  blue-prints."  Whichever  recipe 
is  adopted — and  a  considerable  range  of  choice  will  be  found  in  this  chapter — the  hints  immediately 
following  are  of  general  application. 

272.  Any   white   paper  will   do   for   sensitizing   that   has   a    hard    finish,   like   that   of  ledger    paper, 
so   as   not  to   absorb   the   chemical   solution. 

To  sensitize  the  paper  dissolve  the  ferric  salt  and  the  ferricyanide  in  water,  separately,  as  they 
are  then  not  sensitive  to  the  action  of  light.  The  solutions  should  be  mixed  and  applied  to  the 
paper  only  in  a  dark  room. 

Although  there  is  the  highest  authority  for  "floating  the  paper  to  be  sensitized  for  two  minutes 
on  the  surface  of  the  liquid,"  yet  the  best  American  practice  is  to  apply  the  solution  with  a  soft 
flat  brush  about  four  inches  wide.  The  main  object  is  to  obtain  an  even  coat,  which  may  usually 


•Published    by    the    D.     Van    Nostrand    Company,    New    York.       t American    Edition    revised    and    published    by    Edward    L. 
Wilson,   New   York. 


98  THEORETICAL    AND    PRACTICAL     GRAPHICS. 

be  secured  by  a  primary  coat  of  horizontal  strokes  followed  by  an  overlay  of  vertical  strokes;  the 
second  coat  applied  before  the  first  dries.  If  necessary,  another  coat  of  diagonal  strokes  may  be 
given  to  secure  evenness.  The  thicker  the  coating  given  the  longer  the  time  required  in  printing. 
A  bowl  or  flat  dish  or  plate  will  be  found  convenient  for  holding  the  small  portion  of  the  solution 
required  for  use  at  any  one  time.  The  chemicals  should  not  get  on  the  back  of  the  sheet. 

Each  sheet,  as  coated,  should  be  set  in  a  dark  place  to  dry,  either  "tacked  to  a  board  by  two 
adjacent  corners,''  or  "hung  on  a  rack  or  over  a  rod,"  or  "placed  in  a  drawer — one  sheet  in  a 
drawer," — varying  instructions,  illustrating  the  quite  general  truth  that  there  are  usually  several 
almost  equally  good  ways  of  doing  a  thing. 

273.  To   copy  a   drawing,   place   the    prepared    paper,   sensitized    side    up,   on    a    drawing  -  board    or 
printing -frame   on   which   there   has   been   fastened,   smoothly,   either  a   felt   pad   or    canton    flannel   cloth. 
The    drawing    is    then    immediately   placed    over    the    first    sheet,   inked    side   up,   and    contact    secured 
between   the  two   by  a  large   sheet   of  plate  glass,   placed   over   all. 

Exposure  in  the  direct  rays  of  the  sun  for  four  or  five  minutes  is  usually  sufficient.  The 
progress  of  the  chemical  action  can  be  observed  by  allowing  a  corner  of  the  paper  to  project  beyond 
the  glass.  It  has  a  grayish  hue  when  sufficiently  exposed. 

If  the  sun's  rays  are  not  direct,  or  if  the  day  is  cloudy,  a  proportionately  longer  time  is  required, 
running  up  in  the  latter  case,  from  minutes  into  hours.  Only  experiment  will  show  whether  one's 
solution  is  "quick"  or  "slow;"  or  the  time  required  by  the  degree  of  cloudiness. 

A  solution  will  print  more  quickly  if  the  amount  of  water  in  it  be  increased,  or  if  more  iron 
is  used;  but  in  the  former  case  the  print  will  not  be  as  dark,  while  in  the  latter  the  results,  as  to 
whiteness  of  lines,  are  not  so  apt  to  be  satisfactory. 

Although  fair  results  can  be  obtained  with  paper  a  month  or  more  after  it  has  been  sensitized, 
yet  they  are  far  more  satisfactory  if  the  paper  is  prepared  each  time  (and  dried)  just  before  using. 

On  taking  the  print  out  of  the  frame  it  should  be  immediately  immersed  and  thoroughly 
washed  in  cold  water  for  from  three  to  ten  minutes,  after  which  it  may  be  dried  in  either  of  the 
ways  previously  suggested. 

If  many  prints  are  being  made,  the  water  should  be  frequently  changed  so  as  not  to  become 
charged  with  the  solution. 

274.  The    entire    process,    while    exceedingly   simple    in    theory,   varies,   as    to   its   results,   with   the 
experience   and  judgment   of   the   manipulator.      To   his   choice  the   decision  is   left  between   the   follow- 
ing   standard    recipes    for    preparing    the    sensitizing    solution.      The    "parts"    given    are    all    by   weight. 
In   every   case  the  potash   should   be   pulverized,   to   facilitate   its   dissolving. 

No.    1.      (From   Le    Genie   Civil.) 

.     Red   Prussiate   of  Potash 8   parts. 

Solution   No. 


\     Re 
I    Wa 


/ater 70  parts. 

f    Citrate   of   Iron    and   Ammonia 10   parts. 

N°-   2-    {    Water 70   parts. 

Filter  the   solutions   separately,   mix   equal   quantities   and   then   filter  again. 

No.    2.      (From   U.   S.   Laboratory   at   Willett's   Point). 

f    Double   Citrate   of   Iron   and   Ammonia 1    ounce. 

Solution  No.   1.    \ 

Water 4   ounces. 


Red   Prussiate    of    Potassium 1    ounce. 

Solution  No.   2.    \    ...  ± 

Water 4   ounces. 


tion.    -j 


BLUE-PRINT    PROCESS.  99 

No.   3.      (Lietze's  Method). 

5   ounces,  avoirdupois,    Red    Prussiate   of  Potash. 
Stock  Solution.    -I 

32    fluid    ounces Water. 

"After  the  red  prussiate  of  potash  has  been  dissolved — which  requires  from  one  to  two  days — 
the  liquid  is  filtered.  This  solution  remains  in  good  condition  for  a  long  time.  Whenever  it  is 
required  to  sensitize  paper,  dissolve,  for  every  two  hundred  and  forty  square  feet  of  paper, 

{1    ounce,   avoirdupois,    Citrate   of  Iron   and   Ammonia, 
4}    fluid    ounces Water, 

and  mix   this   with   an   equal   volume   of  the   stock   solution. 

The  reason  for  making  a  stock  solution  of  the  red  prussiate  of  potash  is,  that  it  takes  a  con- 
siderable time  to  dissolve  and  because  it  must  be  filtered.  There  are  many  impurities  in  this 
chemical  which  can  be  removed  by  filtering.  Without  filtering,  the  solution  will  not  look  clear. 
The  reason  for  making  no  stock  solution  of  the  ferric  citrate  of  ammonia  is  that  such  solution  soon 
becomes  moldy  and  unfit  for  use.  This  ferric  salt  is  brought  into  the  market  in  a  very  pure  state, 
and  does  not  need  to  be  filtered  after  being  dissolved.  It  dissolves  very  rapidly.  In  the  solid 
form  it  may  be  preserved  for  an  unlimited  time,  if  kept  in  a  well -stoppered  bottle  and  protected 
against  the  moisture  of  the  atmosphere.  A  solution  of  this  salt,  or  a  mixture  of  it  with  the  solution 
of  red  prussiate  of  potash,  will  remain  in  a  serviceable  condition  for  a  number  of  days,  but  it  will 
spoil,  sooner  or  later,  according  to  atmospheric  conditions.  .  .  .  Four  ounces  of  sensitizing  solution, 
for  blue  prints,  are  amply  sufficient  for  coating  one  hundred  square  feet  of  paper,  and  cost  about 
six  cents." 

For  copying  tracings  in  blue  lines  or  black  on  a  white  ground,  one  may  either  employ  the 
recipes  given  in  Lietze's  and  Pettit's  work,  or  obtain  paper  already  sensitized,  from  the  leading  dealers 
in  draughtsmen's  supplies.  The  latter  course  has  become  quite  as  economical,  also,  for  the  ordinary  blue- 
print, as  the  preparing  of  one's  own  supply. 

For  copying  a  drawing  in  any  desired  color  the  following  method,  known  as  Tilhet's,  is  said  to 
give  good  results:  "The  paper  on  which  the  copy  is  to  appear  is  first  dipped  in  a  bath  con- 
sisting of  30  parts  of  white  soap,  30  parts  of  alum,  40  parts  of  English  glue,  10  parts  of 
albumen,  2  parts  of  glacial  acetic  acid,  10  parts  of  alcohol  of  60°,  and  500  parts  of  water.  It  is 
afterward  put  into  a  second  bath,  which  contains  50  parts  of  burnt  umber  ground  in  alcohol, 
20  parts  of  lampblack,  10  parts  of  English  glue,  and  10  parts  of  bichromate  of  potash  in  500  parts 
of  water.  They  are  now  sensitive  to  light,  and  must,  therefore,  be  preserved  in  the  dark.  In 
preparing  paper  to  make  the  positive  print  another  bath  is  made  just  like  the  first  one,  except  that 
lampblack  is  substituted  for  the  burnt  umber.  To  obtain  colored  positives  the  black  is  replaced  by 
some  red,  blue  or  other  pigment. 

In  making  the  copy  the  drawing  to  be  copied  is  put  in  a  photographic  printing  frame,  and  the 
negative  paper  laid  on  it,  and  then  exposed  in  the  usual  manner.  In  clear  weather  an  illumination 
of  two  minutes  will  suffice.  After  the  exposure  the  negative  is  put  in  water  to  develop  it,  and  the 
drawing  will  appear  in  white  on  a  dark  ground;  in  other  words,  it  is  a  negative  or  reversed  picture- 
The  paper  is  then  dried  and  a  positive  made  from  it  by  placing  it  on  the  glass  of  a  printing- 
frame,  and  laying  the  positive  paper  upon  it  and  exposing  as  before.  After  placing  the  frame  in 
the  sun  for  two  minutes  the  positive  is  taken  out  and  put  in  water.  The  black  dissolves  off  without 
the  necessity  of  moving  back  and  forth." 


100  THEORETICAL    AND    PRACTICAL    GRAPHICS. 


PHOTO -AND     OTHER     PROCESSES. 

275.  If  a   drawing   is   to   be   reproduced   on   a   different   scale   from   that   of   the   original,   some   one 
of    the    processes    which    admits    of    the    use    of   the    camera    is    usually    employed.      Those    of   most 
importance  to   the   draughtsman   are    (1)    wood    engraving;    (2)    the    "wax    process"    or    cerography;    (3) 
lithography,  and   (4)   the   various    methods    in    which   the    photographic    negative   is    made    on    a    film    of 
gelatine    which    is    then    used    directly — to    print    from,   or    indirectly — in    obtaining    a   metal    plate    from 
which   the   impressions   are  taken. 

In  the  first  three  named  above  the  use  of  the  camera  is  not  invariably  an  element  of  the 
process. 

All  under  the  fourth  head  are  essentially  photo -processes  and  their  already  large  number  is 
constantly  increasing.  Among  them  may  be  mentioned  photogravure,  collotype,  phototype,  autotype,  photo- 
glyph, albertype,  heliotype,  and  heliogravure. 

WOOD     ENGRAVING. 

276.  There   is   probably   no   process   that   surpasses   the    best    work    of   skilled    engravers    on    wood. 
This   statement   will   be   sustained   by   a  glance   at   Figs.    14,    15,   20-24,   134,  136,   and  those  illustrating 
mathematical    surfaces,   in    the    next    chapter.      Its    expensiveness,   and   the   time    required    to    make    an 
illustration   by   this   method,   are  its   only   disadvantages. 

Although  the  camera  is  often  employed  to  transfer  the  drawing  to  the  boxwood  block  in  which 
the  lines  are  to  be  cut,  yet  the  original  drawing  is  quite  as  frequently  made  in  reverse,  directly  on 
the  block,  by  a  professional  draughtsman  who  is  supposed  to  have  at  his  disposal  either  the  object 
to  be  drawn  or  a  photograph  or  drawing  thereof.  The  outlines  are  pencilled  on  the  block,  and  the 
shades  and  shadows  given  in  brush  tints  of  India  ink,  re-enforced,  in  some  cases,  by  the  pencil,  for 
the  deepest  shadows. 

The  "high  lights"  are  brought  out  by  Chinese  white.  A  medium  wash  of  the  latter  is  also 
usually  spread  upon  the  block  as  a  general  preliminary  to  outlining  and  shading. 

The  task  of  the  engraver  is  to  reproduce  faithfully  the  most  delicate  as  well  as  the  strongest 
effects  obtained  on  the  block  with  pencil  and  brush,  cutting  away  all  that  is  not  to  appear  in  black 
in  the  print.  The  finished  block  may  then  be  used  to  print  from  directly,  or  an  electrotype  block 
can  be  obtained  from  it  which  will  stand  a  large  number  of  impressions  much  better  than  the  wood. 

CEROGRAPHY. 

277.  For   map-making,    illustrations   of   machinery,   geometrical    diagrams    and    all    work   mainly   in 
straight    lines    or    simple    curves,   and    not    involving    too    delicate  gradations,   the  cerographic   or  "wax 
process"   is   much   employed.      For   clearness  it   is   scarcely   surpassed   by   steel    engraving.      Figures    36, 
90  and    107   are  good   specimens   of   the   effects   obtainable    by   this    method.      The    successive    steps    in 
the   process   are   (a)   the  laying   of  a  thin,   even   coat   of  wax   over  a   copper   plate;    (b)   the  transfer  of 
the    drawing   to    the    surface   of  the   wax,   either    by   tracing    or — more   generally  —  by   photography;    (c) 
the   re -drawing  or   rather  the  cutting   of  these  lines   in  the  wax,  the  stylus   removing  the  latter  to  the 
surface    of   the    copper;    (d)    the    taking    of   an    electrotype    from    the    plate    and    wax,   the    deposit   of 
copper   filling  in   the   lines   from   which   the  wax   was   removed. 

Although  in  the  preparation  of  the  original  drawing  the  lines  may  preferably  be  inked,  yet  it  is 
not  absolutely  necessary,  provided  a  pencil  of  medium  grade  be  employed. 


LITHOGRAPHY.— PHOTO-ENGRAVING.  101 

Any  letters  desired  on  the  final  plate  may  be  also  pencilled  in  their  proper  places,  as  the 
engraver  makes  them  on  the  wax  with  type. 

A  surface  on  which  section -lining  or  cross-hatching  is  desired  may  have  that  fact  indicated  upon 
it  in  writing,  the  direction  and  number  of  lines  to  the  inch  being  given.  Such  work  is  then  done 
with  a  ruling  machine. 

Errors  may  readily  be  corrected,  as  the  surface  of  the  wax  may  be  made  smooth,  for  recutting, 
by  passing  a  hot  iron  over  it. 

LITHOGRAPHY.  —  PHOTO  -  LITHOGRAPHY.  —  CHROMO  -  LITHOGRAPHY. 

278.  For  lithographic  processes  a   fine-grained,   imported   limestone    is    used.      The    drawing  is   made 
with    a    greasy   ink  —  known    as    "lithographic" — upon    a    specially    prepared    paper,   from    which    it   is 
transferred    under    pressure,   to  the   surface    of   the   stone.      The   un- inked   parts   of   the   stone  are  kept 
thoroughly   moistened   with   water,    which    prevents   the   printer's   ink    (owing    to    the    grease    which    the 
latter  contains)   from   adhering   to   any   portion   except  that  from  which   the  impressions   are   desired. 

Photo -lithography  is  simply  lithography,  with  the  camera  as  an  adjunct.  The  positive  might  be 
made  directly  upon  the  surface  of  the  stone  by  coating  the  latter  with  a  sensitizing  solution;  but, 
in  general,  for  convenience,  a  sensitized  gelatine  film  is  exposed  under  the  negative,  and  by  subsequent 
treatment  gives  an  image  in  relief  which,  after  inking,  can  be  transferred  to  the  surface  of  the  stone 
as  in  the  ordinary  process. 

Chromo- lithography,  or  lithography  in  colors,  has  been  a  very  expensive  process,  owing  to  its 
requiring  a  separate  stone  for  each  color.  Recent  inventions  render  it  probable  that  it  will  be  much 
simplified,  and  the  expense  correspondingly  reduced.  The  details  of  manipulation  are  closely  analogous 
to  those  of  ink  prints. 

When  colored  plates  are  wanted,  in  which  delicate  gradations  shall  be  indicated,  chromo  -  litho- 
graphy may  preferably  be  adopted;  although  "half-tones,"  with  colored  inks,  give  a  scarcely  less 
pleasing  effect,  as  illustrated  by  Pigs.  7-10,  Plate  II.  But  for  simple  line-work,  in  two  or  more 
colors,  one  may  preferably  employ  either  cerography  or  photo -engraving,  each  of  which  has  not  only 
an  advantage,  as  to  expense,  over  any  lithographic  process,  but  also  this  in  addition — that  the 
blocks  can  be  used  by  any  printer;  whereas  lithographing  establishments  necessarily  not  only  prepare 
the  stone  but  also  do  the  printing. 

PHOTO  -  ENGRAVING. —  PHOTO  -  ZINCOGRAPHY. 

279.  In   this   popular  and  rapid   process   a  sensitized   solution   is   spread   upon    a    smooth    sheet    of 
zinc,  and   over  this  the   photographic   negative  is   placed.      Where  not  acted   on  by  the  light  the  coat- 
ing   remains    soluble    and    is    washed    away,   exposing    the    metal,   which    is    then   further  acted   on   by 
acids  to  give  more  relief  to   the  remaining   portions. 

Except  as  described  in  Art,  281  this  process  is  only  adapted  to  inked  work  in  lines  or  dots, 
which  is  reproduced  faithfully,  to  the  smallest  detail.  Among  the  best  photo  -  engravings  in  this  book 
are  Figs.  10  and  11,  50,  79  and  80. 

280.  The   following  instructions   for  the   preparation   of  drawings,   for  reproduction   by   this  process, 
are  those   of   the   American    Society   of    Mechanical    Engineers    as    to   the   illustration    of   papers    by  its 
members,   and   are,   in  general,   such   as   all  the   engraving  companies   furnish   on  application. 

"All  lines,  letters  and  figures  must  be  perfectly  black  on  a  white  ground.  Blue  prints  are  not 
available,  and  red  figures  and  lines  will  not  appear.  The  smoother  the  paper,  and  the  blacker  the 
ink,  the  better  are  the  results.  Tracing -cloth  or  paper  answers  very  well,  but  rough  paper — even 


102  THEORETICAL    AND    PRACTICAL     GRAPHICS. 

Whatman's — gives  bad  lines.  India  ink,  ground  or  in  solution,  should  be  used;  and  the  best  lines 
are  made  on  Bristol  board,  or  its  equivalent  with  an  enameled  surface.  Brush  work,  in  tint  or 
grading,  unfits  a  drawing  for  immediate  use,  since  only  line  work  can  be  photographed.  Hatching 
for  sections  need  not  be  completed  in  the  originals,  as  it  can  be  done  easily  by  machine  on  the 
block.  If  draughtsmen  will  indicate  their  sections  unmistakably,  they  will  be  properly  lined,  and 
tints  and  shadows  will  be  similarly  treated. 

The  best  results  may  be  expected  by  using  an  original  twice  the  height  and  width  of  the 
proposed  block.  The  reduction  can  be  greater,  provided  care  has  been  taken  to  have  the  lines  far 
enough  apart,  so  as  not  to  mass  them  together.  Lines  in  the  plate  may  run  from  70  to  100  to 
the  inch,  and  there  should  be  but  half  as  many  in  a  drawing  which  is  to  be  reduced  one -half; 
other  reductions  will  be  in  like  proportion. 

Draughtsmen  may  use  photographic  prints  from  the  objects  if  they  will  go  over  with  a  carbon 
ink  all  the  lines  which  they  wish  reproduced.  The  photographic  color  can  be  bleached  away  by 
flowing  a  solution  of  bi- chloride  of  mercury  in  alcohol  over  the  print,  leaving  the  pen  lines  only. 
Use  half  an  ounce  of  the  salt  to  a  pint  of  alcohol. 

Finally,  lettering  and  figures  are  most  satisfactorily  printed  from  type.  Draughtsmen's  best  efforts 
are  usually  thus  excelled.  Such  letters  and  figures  had  therefore  best  be  left  in  pencil  on  the 
drawings,  so  they  will  not  photograph  but  may  serve  to  show  what  type  should  be  inserted." 

To  the  above  hints  should  be  added  a  caution  as  to  the  use  of  the  rubber.  It  is  likely  to 
diminish  the  intensity  of  lines  already  made  and  to  affect  their  sharpness;  also  to  make  it  more 
difficult  to  draw  clear-cut  lines  wherever  it  has  been  used. 

It  may  be  remarked  with  regard  to  the  foregoing  instructions  that  they  aim  at  securing  that 
uniformity,  as  to  general  appearance,  which  is  usually  quite  an  object  in  illustration.  But  where  the 
preservation  of  the  individuality  and  general  characteristics  of  one's  work  is  of  any  importance  what- 
ever, the  draughtsman  is  advised  to  letter  his  own  drawings  and  in  fact  finish  them  entirely,  himself, 
with,  perhaps,  the  single  exception  of  section  -  lining,  which  may  be  quickly  done  by  means  of  Day's 
Rapid  Shading  Mediums  or  by  other  technical  processes. 

281.  Half  Tones.  Photo -zincography  may  be  employed  for  reproducing  delicate  gradations  of  light 
and  shade,  by  breaking  up  the  latter  when  making  the  photographic  negative.  The  result  is  called 
a  half  tone,  and  it  is  one  of  the  favorite  processes  for  high-grade  illustration.  Figs.  95  and  130 
illustrate  the  effects  it  gives.  On  close  inspection  a  series  of  fine  dots  in  regular  order  will  be  noticed, 
or  else  a  net- work,  so  that  no  tone  exists  unbroken,  but  all  have  more  or  less  white  in  them. 

The  methods  of  breaking  up  a  tone  are  very  numerous.  The  first  patent  dates  back  to  1852. 
The  principle  is  practically  the  same  in  all,  viz.,  between  the  object  to  be  photographed  and  the 
plate  on  which  the  negative  is  to  be  made  there  is  interposed  a  "screen"  or  sheet  of  thin  glass,  on 
which  the  desired  mesh  has  been  previously  photographed. 

In  the  making  of  the  "screen"  lies  the  main  difference  between  the  variously  -  named  methods. 
In  Meissenbach's  method,  by  which  Figs.  95  and  130  were  made,  a  photograph  is  first  taken,  on  the 
"screen,"  of  a  pane  of  clear  glass  in  which  a  system  of  parallel  lines — one  hundred  and  fifty  to  the 
inch  —  has  been  cut  with  a  diamond.  The  ruled  glass  is  then  turned  at  right  angles  to  its  first 
position  and  its  lines  photographed  on  the  screen  over  the  first  set,  the  times  of  exposure  differing 
slightly  in  the  two  cases,  being  generally  about  as  2  to  3. 

This  process  is  well  adapted  to  the  reproduction  of  "wash"  or  brush -tinted  drawings,  photo- 
graphs, etc.  The  object  to  be  represented,  if  small,  may  preferably  be  furnished  to  the  engraving 
company  and  they  will  photograph  it  direct. 


PHOTOGRAPHIC    ILLUSTRATIVE    PR  0  C  E  S  S  E 


GELATINE     FILM     PHOTO- PROCESSES. 

282.  As  stated  in  Art.  275,  in  which  a  few  of  the  above  processes  are  named,  a  gelatine  film 
may  be  employed,  either  as  an  adjunct  in  a  method  resulting  in  a  metal  block,  or  to  print  from 
directly;  in  the  latter  case  the  prints  must  be  made,  on  special  paper,  by  the  company  preparing 
the  film.  In  the  composition  and  manipulation  of  the  film  lies  the  main  difference  between  otherwise 
closely  analogous  processes.  For  any  of  them  the  company  should  be  supplied  with  either  the  original 
object  or  a  good  drawing  or  photographic  negative  thereof. 

Not  to  unduly  prolong  this  chapter — which  any  sharp  distinction  between  the  various  methods 
would  involve,  yet  to  give  an  idea  of  the  general  principles  of  a  gelatine  process  we  may  conclude 
with  the  details  of  the  preparation  of  a  heliotype  plate,  given  in  the  language  of  the  circular  of  a 
leading  illustrating  company.  Figs.  1 — 5  of  Plate  II  illustrate  the  effect  obtained  by  it. 

"Ordinary  cooking  gelatine  forms  the  basis  of  the  positive  plate,  the  other  ingredients  being  bichro- 
mate of  potash  and  chrome  alum.  It  is  a  pecularity  of  gelatine,  in  its  normal  condition,  that  it  will 
absorb  cold  water,  and  swell  or  expand  under  its  influence,  but  that  it  will  dissolve  in  hot  water.  In  the 
preparation  of  the  plate,  therefore,  the  three  ingredients  just  named,  being  combined  in  suitable  propor- 
tions, are  dissolved  in  hot  water,  and  the  solution  is  poured  upon  a  level  plate  of  glass  or  metal,  and 
left  there  to  dry.  When  dry  it  is  about  as  thick  as  an  ordinary  sheet  of  parchment,  and  is  stripped 
from  the  drying -plate,  and  placed  in  contact  with  the  previously  -  prepared  negative,  and  the  two 
together  are  exposed  to  the  light.  The  presence  of  the  bichromate  of  potash  renders  the  gelatine 
sheet  sensitive  to  the  action  of  light;  and  wherever  light  reaches  it,  the  plate,  which  was  at  first 
gelatinous  or  absorbent  of  water,  becomes  leathery  or  waterproof.  In  other  words,  wherever  light 
reaches  the  plate,  it  produces  in  it  a  change  similar  to  that  which  tanning  produces  upon  hides  in 
converting  them  into  leather.  Now  it  must  be  understood  that  the  negative  is  made  up  of  trans- 
parent parts  and  opaque  parts;  the  transparent  parts  admitting  the  passage  of  light  through  them, 
and  the  opaque  parts  excluding  it.  When  the  gelatine  plate  and  the  negative  are  placed  in  contact, 
they  are  exposed  to  light  with  the  negative  uppermost,  so  that  the  light  acts  through  the  translucent 
portions,  and  waterproofs  the  gelatine  underneath  them;  while  the  opaque  portions  of  the  negative 
shield  the  gelatine  underneath  them  from  the  light,  and  consequently  those  parts  of  the  plate  remain 
unaltered  in  character.  The  result  is  a  thin,  flexible  sheet  of  gelatine  of  which  a  portion  is  water- 
proofed, and  the  other  portion  is  absorbent  of  water,  the  waterproofed  portion  being  the  image  which 
we  wish  to  reproduce.  Now  we  all  know  the  repulsion  which  exists  between  water  and  any  form 
of  grease.  Printer's  ink  is  merely  grease  united  with  coloring- matter.  It  follows,  that  our  gelatine 
sheet,  having  water  applied  to  it,  will  absorb  the  water  in  its  unchanged  parts;  and,  if  ink  is  then 
rolled  over  it,  the  ink  will  adhere  only  to  the  waterproofed  or  altered  parts.  This  flexible  sheet  of 
gelatine,  then,  prepared  as  we  have  seen,  and  having  had  the  image  impressed  upon  it,  becomes  the 
heliotype  plate,  capable  of  being  attached  to  the  bed  of  an  ordinary  printing-press,  and  printed  in  the 
ordinary  manner.  Of  course,  such  a  sheet  must  have  a  solid  base  given  to  it,  which  will  hold  it 
firmly  on  the  bed  of  the  press  while  printing.  This  is  accomplished  by  uniting  it,  under  water,  with 
a  metallic  plate,  exhausting  the  air  between  the  two  surfaces,  and  attaching  them  by  atmospheric 
pressure.  The  plate,  with  the  printing  surface  of  gelatine  attached,  is  then  placed  on  an  ordinary 
platen  printing-press,  and  inked  up  with  ordinary  ink.  A  mask  of  paper  is  used  to  secure  white 
margins  for  the  prints;  and  the  impression  is  then  made,  and  is  ready  for  issue." 


MECHANICAL    DRAWINGS.  — WORKING    DRAWINGS.  131 


CHAPTER    X. 

PROJECTIONS     AND     INTERSECTIONS     BY     THE     THIRD-ANGLE     METHOD.— THE     DEVELOPMENT     OP 

SURFACES     FOR    SHEET    METAL    PATTERN     MAKING.— UPPER    CHORD    POST-CONNECTION 

OF    A    RAILROAD    BRIDGE—SPUR    GEAR.— HELICAL    SPRINGS.— BOLTS, 

SCREWS    AND    NUTS,     WITH    TABLES. 

383.  The   mechanical   drawings   preliminary   to   the   construction   of  machinery,  blast  furnaces,  stone 
arches,   buildings,   and,   in   fact,  all  architectural   and   engineering   projects,  are  made   in   accordance  with 
the   principles   of  Descriptive   Geometry.      When   fully   dimensioned   they  are   called  working  drawings. 

The  object  to  be  represented  is  supposed  to  be  placed  in  either  the  first  or  the  third  of  the 
four  angles  formed  by  the  intersection  of  a  horizontal  plane,  H,  with  a  vertical  plane,  V.  (Fig.  228). 

The  representations  of  the  object  upon  the  planes  are,  in  mathematical  language,  projections,  and 
are  obtained  by  drawing  perpendiculars  to  the  planes  H  and  V  from  the  various  points  of  the 
object,  the  point  of  intersection  of  each  such  projecting  line  with  a  plane  giving  a  projection  of  the 
original  point.  Such  drawings  are,  obviously,  not  "views"  in  the  ordinary  sense,  as  they  lack  the 
perspective  effect  which  is  involved  in  having  the  point  of  sight  at  a  finite  distance;  yet  in  ordinary 
parlance  the  terms  top  view,  horizontal  projection  and  plan  are  used  synonymously ;  as  are  front  view 
and  front  elevation  with  vertical  projection,  and  side  elevation  with  profile  view,  the  latter  on  a  plane 
perpendicular  to  both  H  and  V,  and  called  the  profile  plane. 

Although  the  words  "plan"  and  " elevation "  are  the  ones  most  frequently  employed  by  architects, 
while  engineers  generally  give  "view"  or  "projection"  the  preference,  no  attempt  at  uniformity  in 
their  use  has  been  made  in  the  following  matter,  the  aim  being  rather,  by  their  occurring  inter- 
changeably, to  familiarize  the  student  equally  with  standard  terminology. 

Until  the  last  decade  of  the  first  century  of  Descriptive  Geometry  (1795-1895)  problems  were 
solved  as  far  as  possible  in  the  first  angle.  As  the  location  of  the  object  in  the  third  angle  —  that 
is,  below  the  horizontal  plane  and  behind  the  vertical  —  results  in  a  grouping  of  the  views  which  is 
in  a  measure  self- interpreting,  the  Third  Angle  Method  is,  however,  to  a  considerable  degree  supplant- 
ing the  other  for  machine-shop  work. 

The  advantageous  grouping  of  the  projections  which  constitutes  the  only  —  though  a  quite  suf- 
ficient— justification  for  giving  it  special  treatment,  is  this:  The  front  view  being  always  the  central 
one  of  the  group,  the  top  view  is  found  at  the  top;  the  view  of  the  right  side  of  the  object  appears 
on  the  right;  of  the  left-hand  side  on  the  left,  etc.  Thus,  in  Fig.  228  (a),  with  the  hollow  block 
BDFS  as  the  object  to  be  represented,  we  have  ades  for  its  horizontal  projection,  c'  d'  e' f  for  its 
vertical  projection,  f"e"s"x"  for  the  side  elevation;  then  on  rotating  the  plane  H  clockwise  on  G.  L. 
into  coincidence  with  V,  and  the  profile  plane  P  about  QR  until  the  projection  /"  e"  s"  x"  reaches 
f'"e'"s'"x'",  we  would  have  that  location  of  the  views  which  has  just  been  described. 

The  lettering  shows  that  each  projection  represents  that  side  of  the  object  which  is  toward  the 
plane  of  projection. 

384.  The  same  grouping  can  be  arrived   at   by  a  different  conception,  which  will,  to  some,  have 
advantages  over  the  other.      It   is  illustrated    by  Fig.  228  (b),    in  which   the  same  object   as  before  is 


132 


THEORETICAL    AND    PRACTICAL    GRAP-HICS. 


supposed  to  be  surrounded  by  a  system  of  mutually -perpendicular  transparent  planes,  or,  in  other 
words,  to  be  in  a  box  having  glass  sides,  and  on  each  side  a  drawing  made  of  what  is  seen  through 
that  side,  excluding  the  idea,  as  before,  of  perspective  view,  and  representing  each  point  by  a  per- 

3Figr-  see. 


pendicular  from  it  to  the  plane.  The  whole  system  of  box  and  planes,  in  the  wood -cut,  is  rotated 
90  °  from  the  position  shown  in  Fig.  228  (a),  bringing  them  into  the  usual  position,  in  which  the 
observer  is  looking  perpendicularly  toward  the  vertical  plane. 


385.  In  Fig.  229  we  may  illustrate  either  the  First  or  the  Third  Angle  method,  as  to  the  top 
view  of  the  object;  ades  in  the  upper  plane  being  the  plan  by  the  latter  method,  and  a1d1el8l 
by  the  former. 


ORTHOGRAPHIC    PROJECTION    OF    SOLIDS. 


133 


Disregarding  Q  TXN  we  have  the  object  and  planes  illustrating  the  first -angle  method  through- 
out, the  lettering  of  each  projection  showing  that  it  represents  the  side  of  the  object  farthest  from  the 
plane,  making  it  the  exact  reverse  of  the  third -angle  system. 

In  the  ordinary  representation  the  same  object  would  be  represented  simply  by  its  three  views  as 
in  Fig.  230.  In  the  elevations  the  short -dash  lines  indicate  the  invisible  edges  of  the  hole. 

The  arcs  show  the  rotation  which  carries  the  profile  view  into  its  proper  place. 


-.  231. 


386.  For  the  sake  of  more  readily  contrasting  the  two  methods  a  group  of  views  is  shown  in 
Fig.  231,  all  above  G.  L.,  illustrating  an  object  by  the  First  Angle  system,  while  all  below  H K 
represents  the  same  object  by  the  Third  Angle  method. 

When  looking  at  Figures  1,  2,  3  and  4  the  observer  queries:  What  is  the  object,  in  space,  whose 
front  is  like  Fig.  1,  top  is  like  Fig.  2,  left  side  is  like  Fig.  3  and  right  side  like  Fig.  4? 

For  the  view  of  the  left  side  he  might  imagine  himself  as  having  been  at  first  between  G  and 
H,  looking  in  the  direction  of  arrow  N,  after  which  both  himself  and  the  object  were  turned,  together 


134  THEORETICAL    AND    PRACTICAL    GRAPHICS. 

to  the  right,  through  a  ninety  -  degree  arc,  when  the  same  side  would  be  presented  to  his  view 
in  Fig.  3.  Similarly,  looking  in  the  direction  of  the  arrow  M,  an  equal  rotation  to  the  left,  as 
indicated  by  the  arcs  1-2,  3-4,  5-6,  etc.,  would  give  in  Fig.  4  the  view  obtained  from  direction 
M.  His  mental  queries  would  then  be  answered  about  as  follows:  Evidently  a  cubical  block  with 
a  rectangular  recess — r'v'd'c' — in  front;  on  the  rear  a  prismatic  projection,  of  thickness  ph  and 
whose  height  equals  that  of  the  cube;  a  short  cylindrical  ring  projecting  from  the  right  face  of  the 
cube;  an  angular  projecting  piece  on  the  left  face. 

In  Fig.  2  the  line  rv  is  in  short  dashes,  as  in  that  view  the  back  plane  of  the  recess  r'v'd'c' 
would  be  invisible.  In  Fig.  4  the  back  plane  of  the  same  recess  is  given  the  letters,  v"d",  of  the 
edge  nearest  the  observer  from  direction  M. 

To  illmtrate  the  third  angle  method  by  Fig.  231  we  ignore  all  above  the  line  H  K.  In  Fig.  5  we 
have  the  same  front  elevation  as  before,  but  above  it  the  view  of  the  top;  below  it  the  view  of  the 
bottom  exactly  as  it  would  appear  were  the  object  held  before  one  as  in  Fig.  5,  then  given  a  ninety- 
degree  turn,  around  a'b',  until  the  under  side  became  the  front  elevation. 

Fig.  7  may  as  readily  be  imagined  to  be  obtained  by  a  shifting  of  the  object  as  by  the  rotation 
of  a  plane  of  projection;  for  by  translating  the  object  to  the  right,  from  its  position  in  Fig.  5,  then 
rotating  it  to  the  left  90°  about  b'n',  its  right  side  would  appear  as  shown. 

387.  For   convenient  reference    a    general    resume   of   terms,   abbreviations   and  instructions   is   next 
presented,   once   for  all,  for  use  in  both   the   Third   Angle  and   First   Angle   methods. 

(1)  H,   V,   P the  horizontal,   vertical  and  profile  planes  of  projection  respectively. 

(2)  H  -  projector the  projecting  line   which   gives  the  horizontal  projection  of  a  point. 

(3)  V- projector the  projecting  line  giving  the   projection   of  a  point  on  V. 

(4)  Projector -plane the   profile  plane  containing  the  projectors  of  a  point. 

(5)  h.  p the   horizontal  projection  or  plan   of  a   point   or   figure. 

(6)  v.  p the   vertical  projection   or   elevation   of  a   point   or   figure. 

(7)  h.  t horizontal   trace,   the  intersection   of  a   line  or  surface   with   H. 

(8)  v.  t vertical  trace,   the  intersection   of  a  line  or  surface   with   V. 

(9)  H  -  traces,  V  -  traces plural   of  horizontal   and   vertical  traces  respectively. 

(10)  G.  L ground  line,    the  line  of  intersection   of  V   and   H. 

(11)  V- parallel a  line  parallel  to  V   and  lying  in   a  given   plane. 

(12)  A  horizontal any  horizontal   line   lying   in   a  given   plane. 

(13)  Line  of  declivity the  steepest  line,   with   respect   to  one  plane,   that  can   lie   in  another  plane. 

(14)  Kabatment revolution   into   H   or  V   about  an   axis   in  such   plane. 

(15)  Counter  -  rabatment  or   revolution  .  restoration   to   original   position. 

388.  For  Problems  relating   solely  to    the    Point,    Line   and  Plane. 

Qiven  lines  should  be  fine,  continuous,  black ;  required  lines  heavy,  continuous,  black  or  red ;  construction  lines  in 
fine,  continuous  red,  or  short -dash  black;  traces  of  an  auxiliary  plane,  or  invisible  traces  of  any  plane,  in  dash -and- 
three  -  dot  lines. • 

For   Problems   relating    to   Solid    Objects. 

(1)  Pencilling.     Exact ;    generally  completed  for  the  whole  drawing  before  any  inking  is  done  ;    the  work  usually  from 
centre  lines,   and   from   the  larger  —  and   nearer  —  parts  of  the  object  to  the  smaller  or  more  remote. 

(2)  Inking  of  the  Object.      Curves  to  be  drawn   before  their  tangents ;  fine  lines  uniform  and  drawn  before   the  shade 
lines;    shade  lines  next  and   with   one  setting  of  the  pen,   to  ensure  uniformity.      On   tapering  shade   lines  see   Art.    111. 

(3)  Shade  Lines.     In  architectural  work  these   would  be  drawn  in   accordance   with   a  given  direction   of  light. 

In  American  machine-shop  practice  the  right-hand  and  lower  edges  of  a  plane  surface  are  made  shade  lines  if  they 
separate  it  from  invisible  surfaces.  Indicate  curvature  by  line-shading  if  not  otherwise  sufficiently  evident.  (See  Fig.  288). 


THE    CONSTRUCTION    AND    FINISH    OF    WORKING    DRAWINGS.       135 


r-  233. 


(4)  Invisible  lines  of  the  object,   black,   invariably,   in   dashes  nearly   one-tenth   of   an  inch  in  length.  __________ 

(5)  Inking   of  lines   other   than   of    the   object.      When    no    colors    are    to    be    employed    the    following    directions    as    to 
kind  of  line   are  those  most  frequently  made.      The  lines   may   preferably   be   drawn   in   the  order  mentioned. 

Centre    lines,   an   alternation   of   dash   and   two    dots.   ----------  -  ----------  _____ 

Dimension   lines,   a  dash  and  dot  alternately,   with   opening  left  for  the  dimension.   ------  _  .  __ 

Extension    lines,    for   dimensions    placed    outside   the   views,    in   dash-and-dot   as   for   a   dimension   line.  _____ 

Ground  line,   (when   it  cannot  be  advantageously   omitted)   a   continuous   heavy  line.   ^^^^_^_^^^_^^^_^_^___ 
Construction  and   other  explanatory  lines  in   short   dashes.    ---------------  -  ----------  -  ---------------- 

(6)  When  using   colors   the   centre,    dimension  and    extension    lines    may   be    fine,    continuous,    red;    or    the    former    may 
be   blue,   if  preferred.      Construction   lines  may   also   be  red,   in   short    dashes  or  in    fine   continuous  lines. 

Instead  of   using  bottled  inks   the  carmine   and    blue    may   preferably   be    taken    directly   from    "Winsor    and    Newton   cakes, 
"moist  colors."      Ink   ground   from  the  cake   is   also  preferable  to  bottled  ink. 

Drawings  of  developable  and   warped  surfaces  are  much   more  effective  if  their  elements   are  drawn  in  some   color. 

(7)  Dimensions  and  Arrow-Tips.     The   dimensions  should  invariably  be  in    black,    printed  free-hand    with    a    writing- 
pen,    and    should    read    in    line    with    the    dimension   line   they   are    on.       On    the   drawing   as   a   whole   the   dimensions   should 
read  either  from    the    bottom    or  right-hand    side.      Fractions    should    have    a    horizontal    dividing    line;    although    there    is 
high   sanction   for  the  omission  of  the  dividing   line,   particularly   in   a  mixed   number. 

Extended  Gothic,  Roman,  Italic  Roman  and  Reinhardt's  form  of  Condensed  Italic  Gothic  are  the  best  and  most 
generally  used  types  for  dimensioning. 

The  arrow-  tips  are  to  be  always  drawn  free-hand,  in  black;  to  touch  the  lines  between  which  they  give  a  distance; 
and  to  make  an  acute  instead  of  a  right  angle  at  their  point. 

389.  Working  drawing  of  a  right  pyramid;  base,  an  equilateral  triangle  0.9"  on  a  side;  altitude,  x. 
Draw  first  the  equilateral  triangle  a  b  c  for  the  plan  of  the  base,  making  its  sides  of  the  pre- 
scribed length.  If  we  make  the  edge  a  b  perpendicular  to 
the  profile  plane,  01,  the  face  vab  will  then  appear  in 
profile  view  as  the  straight  line  v"b".  Being  a  right  pyra- 
mid, with  a  regular  base,  we  shall  find  v,  the  plan  of  the 
vertex,  equally  distant  from  a,  b  and  c;  and  v  a,  vb,  vc 
for  the  plans  of  the  edges. 

Parallel  to  G.  L.  and  at  a  distance  apart  equal  to  the 
assigned  height,  x,  draw  mv"  and  nc"  as  upper  and  lower 
limits  of  the  front  and  side  elevations;  then,  as  the  h.  p. 
and  v.  p.  of  a  point  are  always  in  the  same  perpendicu- 
lar to  G.  L.,  we  project  v,  a,  b  and  c  to  their  respective 
levels  by  the  construction  lines  shown,  obtaining  v'.a'b'c' 
for  the  front  elevation. 

Projectors  to  the  profile  plane   from  the    points  of  the  plan  give  1,  2,  3,  -which  are  then  carried, 
in  arcs   about   0,   to   L,  5,   4,  and   projected   to   their  proper  levels,   giving  the  side  elevation,  v"b"c". 
As  the  actual  length  of   an  edge  is  not  shown    in    either    of   the    three    views,   we    employ  the    fol- 

lowing  construction  to  ascertain  it:  Draw  vvl  perpendicular  to 
vb,  and  make  it  equal  to  a;  vtb  is  then  the  real  length  of 
the  edge,  shown  by  rabatment  about  vb. 

The  development  of  the  pyramid  (Fig.  233)  may  be  obtained 
by  drawing  an  arc  ABCA1  of  radius  =  t>,  b  (the  true  length 
of  edge,  from  Fig.  232)  and  on  it  laying  off  the  chords  AS, 

^,      ' 

BO,  CAl   equal    to    a  b,    be,    ca    of   the    plan;    then    V-ABCAl 
is    the    plane    area    which,   folded    on    VC  and    VB,    would  give   a  model  of  the  pyramid  represented. 


333- 


136 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


•-  23-3;- 


390.  Working  drawing  of  a  semi -cylindrical  pipe:    outer   diameter,   x;    inner    diameter,  y;    height,  z. 
For  the  plan  draw   concentric   semi -circles   aed  and   bsc,   of    diameters  x  and  y  respectively,  join- 
ing   their    extremities    by  straight    lines   a  b,   c  d.      At  a    distance 

apart  of  z  inches  draw  the  upper  and  lower  limits  of  the 
elevations,  and  project  to  these  levels  from  the  points  of  the 
plan. 

In  the  side  view  the  thickness  of  the  shell  of  the  cylinder 
is  shown  by  the  distance  between  e"f"  and  s"t" — the  latter  so 
drawn  as  to  indicate  an  invisible  limit  or  line  of  the  object. 

The  line  shading  would  usually  be  omitted,  the  shade  lines 
generally  sufficing  to  convey  a  clear  idea  of  the  form. 

391.  Half  of  a  hollow,  hexagonal  prism.      In   a  semi -circle   of 
diameter   a  d  step   off  the   radius   three  times   as   a  chord,  giving 
the   vertices   of  the  plan  abed  of  the  outer  surface.     Parallel  to 
b  c,   and   at   a   distance   from   it   equal    to    the    assigned    thickness 
of   the    prism,    draw    ef,    terminating    it    on    lines    (not    shown) 


•.  335.  drawn   through   b    and   c  at   60  °    to   a  d.      From   e  and  /   draw   e  h 

and  fg,  parallel  respectively  to  ab  and  c  d.  Drawing  a' c"  and 
m't"  as  upper  and  lower  limits,  project  to  them  as  in  preceding 
problems  for  the  front  and  side  elevations. 

392.     Working    drawing    of    a    hollow,    prismatic    block,    standing 
obliquely    to    the    vertical    and   profile   planes. 

Let  the  block  be  2"x3"xl"  outside,  with  a  square  open- 
ing 1 "  x  1 "  x  1 "  through  it  in  the  direction  of  its  thickness. 
Assuming  that  it  has  been  required  that  the  two -inch  edges 
should  be  vertical,  we  first  draw,  in  Fig.  236,  the  plan  asxb, 
3"xl",  011  a  scale  of  1:2.  The  inch -wide  opening  through  the 
centre  is  indicated  by  the  short -dash  lines. 
B'  n"  t"  For  the  elevations  the  upper  and  lower  limits  are  drawn  2" 

apart,   and   a,  b,  s,  x,   etc.,    projected  to   them.      The 

elevations   of  the  opening  are  between  levels  m'm" 

and   k'k",  one   inch    apart   and    equi- distant    from 

the  upper  and   lower   outlines  of  the   views.      The 

dotted    construction    lines    and    the    lettering    will 

enable    the    student    to    recognize   the   three    views 

of  any   point   without   difficulty. 

393.     In  Fig.  237  we  have  the  same  object  as 

that  illustrated  by   Fig.  236,   but    now    represented 

as   cut  by  a  vertical   plane   whose   horizontal  trace 

is  v  y.      The   parts   of   the   block  that  are  actually 

cut  by   the  plane    are    shown    in    section -lines    in 

the    elevations.      This    is    done    here    and   in  some 

later    examples     merely    to    aid    the    beginner    in 

understanding  the  views;    but,   in  engineering  prac- 
tice,   section-lining    is    rarely    done    on    views    not   perpendicular    to    the   section  plane. 


PROJECTION    OF    SO  LIDS.— WORKING    DRAWINGS. 


137 


is     customary     to     omit     the 


-.   237-. 


*./. j 

Ifa- 


394.  Suppression    of    the    ground    line.        In     machine     drawing    it 
ground  line,  since  the  forms  of  the  various  views  — 

which  alone  concern  us — are  independent  of  the 
distance  of  the  object  from  an  imaginary  horizon- 
tal or  vertical  plane.  We  have  only  to  remember 
that  all  elevations  of  a  point  are  at  the  same 
level;  and  that  if  a  ground  line  or  trace  of  any 
vertical  plane  is  wanted,  it  will  be  perpendicular 
to  the  line  joining  the  plan  of  a  point  with  its 
projection  on  such  vertical  plane.  (Art.  286.) 

395.  Sections.     Sectional  views.    Although  earlier 
defined  (Art.  70),  a  re-statement  of  the  distinction 
between  these  terms  may  well  precede  problems  in 
which  they  will  be  so  frequently  employed. 

When  a  plane  cuts  a  solid,  that  portion  of  the 
latter  which  comes  in  actual  contact  with  the  cutting 
plane  is  called  the  section. 

A  sectional  view  is  a  view  perpendicular  to  the  cutting  plane,  and  showing  not  only  the  section  but  also 
the  object  itself  as  if  seen  through 

the    plane.        When    the     cutting  „„  SECTIONAL  VIEW 

plane  is  vertical  such  a  view  is 
called  a  sectional  elevation;  when 
horizontal,  a  sectional  plan. 

396.  Working    drawing    of    a 
regular,  pentagonal  pyramid,   hollow, 
truncated  by  an  oblique  plane;    also 
the  development,  or  "pattern,"  of  the 
outer  surface  below  the  cutting  plane. 
For  data  take  the   altitude   at   2"; 
inclination    of   faces,   0°    (meaning 
any    arbitrary    angle);     inclination 
of    section    plane,    30°;     distance 
between   inner  and  outer  faces   of 
pyramid,  \" . 

(1)  Locate  v  and  r'  (Fig.  238) 
for  the  plan  and  elevation  of  the 
vertex,  taking  them  sufficiently 
apart  to  avoid  the  overlapping  of 
one  view  upon  the  other.  Through 
v  draw  the  horizontal  line  S  T, 
regarding  it  not  only  as  a  centre 
line  for  the  plan  but  also  as  the 
h.  t.  of  a  central,  vertical,  refer- 
ence plane,  parallel  to  the  ordi- 
nary vertical  plane  of  projection. 


I 


-V/ ;/--/--/-/-•— -i^ 


138 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


SECTIONAL  VIEW 


PLAN 


(The   student   should   note  that  for  convenient  reference   Fig.  238   is   repeated   on  this  page.) 

On  the  vertical  line  vv'  (at  first  indefinite  in  length)  lay  off  v' s'  equal  to  2",  for  the  altitude 
(and  axis)  of  the  pyramid,  and  through  a'  draw  an  indefinite  horizontal  line,  which  will  contain  the 
v.  p.  of  the  base,  in  both  front  and  side  views. 

Draw  v'b'  at  6°  to  the  horizontal.  It  will  represent  the  v.  p.  of  an  outer  face  of  the  pyramid, 
and  b '  will  be  the  v.  p.  of  the  edge  a  b  of  the  base.  The  base  a  b  c  d  e  is  then  a  regular  pentagon 
circumscribed  about  a  circle  of  centre  v  and  radius  vi  =  s'b'.  Since  the  angle  avb  is  72°  (Art.  92) 
we  get  a  starting  corner,  a  or  b,  by  drawing  v  a  or  v  b  at  36  °  to  S  T,  to  intercept  the  vertical  through 
b'.  The  plans  of  the  edges  of  the  pyramid  are  then  v  a,  vb,  v  c,  vd  and  v  e.  Project  d  to  d'  and 
draw  v' d'  for  the  elevation  of 
v  d ;  similarly  for  v  e  and  v  c, 
which  happen  in  this  case  to  co- 
incide in  vertical  projection. 

For  the  inner  surface  of  the 
pyramid,  whose  faces  are  at  a 
perpendicular  distance  of  \"  from 
the  outer,  begin  by  drawing  g' I' 
parallel  to  and  \"  from  the  face 
projected  in  b' v' ;  this  will  cut 
the  axis  at  a  point  t'  which  will 
be  the  vertex  of  the  inner  sur- 
face, and  g' t'  will  represent  the 
elevation  of  the  inner  face  that  is 
parallel  to  the  face  a  v  b  —  v '  b ' ; 
while  gh,  vertically  above  g'  and 
included  between  va  and  vb,  will 
be  the  plan  of  the  lower  edge  of 
this  face.  Complete  the  pentagon 
gh  —  k  for  the  plan  of  the  inner 
base;  project  the  corners  to  b' d' 
and  join  with  t'  to  get  the  ele- 
vations of  the  interior  edges. 

The  section.  In  our  figure  let 
G'  H '  be  the  section  plane,  sit- 
uated perpendicular  to  the  ver- 
tical plane  and  inclined  30°  to 
the  horizontal.  It  intersects  v' d'  in  p',  which  projects  upon  vd  at  p.  Similarly,  since  G' H'  cuts 
the  edges  v' c'  and  v' e'  at  points  projected  in  o',  we  project  from  the  latter  to  v  c  and  ve,  obtaining 
o  and  q.  A  like  construction  gives  m  and  n.  The  polygon  mnopq  is  then  the  plan  of  the  outer 
boundary  of  the  section. 

The  inner  edge  g' t'  is  cut  by  the  section  plane  at  I',  which  projects  to  both  v  h  and  v  g,  giving 
the  parallel  to  mn  through  I.  The  inner  boundary  of  the  section  may  then  be  completed  either 
by  determining  all  its  vertices  in  the  same  way  or  on  the  principle  that  its  sides  will  be  parallel  to 
those  of  the  outer  polygon,  since  any  two  planes  are  cut  by  a  third  in  parallel  lines. 

The   line  m' p'  is   the   vertical   projection   of  the   entire   section. 


PROJECTION    OF    SOLIDS.  — WORKING    DRAWINGS.  139 

(2)  The  side  elevation.       This    might    be   obtained   exactly   as   in   the   five   preceding  figures,   that  is, 
by    actually    locating    the    side    vertical,   or   profile,   plane,   projecting    upon    it    and    rotating   through   an 
arc    of   90°.       In    engineering   practice,    however,    the    method    now  to    be    described    is    in   Jar    more  general 
use.       It    does    not    do    away    with    the    profile    plane,   on    the    contrary    presupposes    its   existence,   but 
instead   of  actually  locating  it  and   drawing  the  arcs   which   so   far  have  kept  the  relation  of  the  views 
constantly  before  the   eye,   it  reaches   the   same   result   in  the   following   manner :     A    vertical    line    S'  T' 
is   drawn   at  some  convenient   distance  to   the  right   of   the    front   elevation ;    the    distance,   from  S  T,  of 
any    point    of   the    plan,   is    then    laid    off   horizontally    from    iS"  T",   at    the    same    height    as    the  front 
elevation    of  the    point.      For,  as    earlier   stated,  S  T  was  to  be  regarded  as  the  horizontal  trace  of   a 
vertical    plane.      Such    plane    would    evidently    cut    a   profile    plane  in    a    vertical    line,  which  we  may 
call    S' T',  and    let    the  S' T'   of   our    figure    represent    it    after    a    ninety -degree   rotation  has  occurred. 
The    distances    of   all    points    of   the    object,   to    either    the  front  or  rear  of   the   vertical  plane  on  S  T 
would,   obviously,   be   now   seen   as   distances   to  the    left    or    right,   respectively,   of   the    trace    S' T',   and 
would  be   directly   transferred   with   the   dividers   to    the    lines    indicating    their    level.       Thus,   e"  is    on 
the  level  of  e',   but  is   to   the  right  of  S' T'   the    same    distance    that    e    is   above  (or,  in    reality,   behind) 
the  plane  ST;    that   is,   e" d"   equals   eit.      Similarly   d"b"   equals   ib;    n"x"  equals  nx. 

It  is  usual,  where  the  object  is  at  all  symmetrical,  to  locate  these  reference  planes  centrally,  so 
that  their  traces,  used  as  indicated,  may  bisect  as  many  lines  as  possible,  to  make  one  setting  of  the 
dividers  do  double  work. 

(3)  True    size    of   the    section.       Sectional    mew.       If    the    section    plane    G' H'    were    rotated    directly 
about    its    trace    on    the    central,   vertical    plane    <Sr  T,   until    parallel    to    the    paper,   it    would    show   the 
section    in' p'  —  mnopq    in    its    true    size;     but  such   a   construction   would   cause   a  confusion   of   lines, 
the    new    figure   overlapping    the    front   elevation.      If,   however,   we    transfer    the    plane    G' H'  —  keeping 
it  parallel  to   its  first  position   during  the   motion  —  to   some  new   position  S"T",  and  then  turn  it  90° 
on    that    line,   we    get    m,lnlolp1ql,   the    desired   view    of  the    section.       The    distances    of   the    vertices 
of    the    section    from   S"  T"  are    derived    from    reference    to    S  T    exactly    as    were    those    in    the    side 
elevation;    that   is,   mlx1  =  mx  =  m"x".     We  thus   see  that   one   central,   vertical,   reference   plane,   ST,    is 
auxiliary   to   the   construction   of  two   important  views ;    <S"  T'   represents   its  intersection   with   the  profile 
or    side    vertical    plane,   while   S"  T"   is    its   (transferred)   trace   upon   the   section  plane    G'H'.      For  the 
remainder    of   the    sectional    view    the   points    are    obtained   exactly   as   above   described   for  the   section; 
thus   c'clel   is   perpendicular   to   S"T";    ettil   equals   eu,   and   clu1   equals   CM. 

(4)  To    determine    the    actual    length    of   the    various    edges.       The    only    edge    of    the    original,    uncut 
pyramid,   that  would   require   no   construction    in   order  to   show    its   true  length,   is    the    extreme    right- 
hand   one,   which  —  being   parallel   to   the   vertical   plane,   as   shown   by   its   plan  v  d  being   horizontal  —  is 
seen   in   elevation   in   its   true   size,   v'd'.      Since,   however,   all  the   edges   of   the    pyramid  are  equal,   we 
may   find   on  v'd'  the   true   length   of  any  portion  of  some   other   edge,   as,  for   example   o'c',  by  taking 
that   part   of  v'd'  which   is   intercepted   between  the   same   horizontals,   viz.:   o'"d'. 

Were  we  compelled  to  find  the  true  length  of  o'c',  oc,  independently  of  any  such  convenient 
relation  as  that  just  indicated,  we  would  apply  one  of  the  methods  fully  illustrated  by  Figs.  183,  184 
and  187,  or  the  following  "  shop "  modification  of  one  of  them :  Parallel  to  the  plan  o  c  draw  a  line 
i/z,  their  distance  apart  to  be  equal  to  the  difference  of  level  of  o'  and  c',  which  difference  may  be 
obtained  from  either  of  the  elevations ;  from  the  plan  o  of  the  higher  end  of  the  line  draw  the 
common  perpendicular  of,  and  join  /  with  c,  obtaining  the  desired  length  fc. 

(5)  To    show    the    exact  form    of   any  fare    of   the   pyramid.       Taking,   for    example,   the    face    ocdp, 
revolve   o  p  about   the   horizontal   edge   c  d  until   it  reaches   the   level  of  the   latter.      The  actual  distance 


140 


THEORETICAL     AM)     PRACTICAL     GRAPHICS. 


of  o  from  c,  and  of  p  from  d  will  be  the  same  after  as  before  this  revolution,  while  the  paths  of  o 
and  p  during  rotation  will  be  projected  in  lines  or  and  ptc,  each  perpendicular  to  cd;  therefore, 
with  c  as  a  centre,  cut  the  perpendicular  or  by  an  arc  of  radius  fc  —  just  ascertained  to  be  the  real 
length  of  or,  and,  similarly,  cut  p  w  by  an  arc  of  radius  dw^p'd'i  join  •/•  with  c,  w  with  d. 
draw  w  r  and  we  have  in  c  d  >r  r  the  form  desired. 

(6)  The    development    of   the    outer    xtirfacc  of   the  truncated  pyramid.     With   any   point    V  as   a   centre 
(Fig.  239)   and   with   radius   equal   to   the   actual   length   of   an   edge    of   the   pyramid    (that   is,   equal   to 
v'd',   Fig.  238)    draw   an   indefinite   arc,   on   which   lay   off   the    chords    D  C,   C  B,   HA,   A  E,   ED,   equal 
respectively  to   the  like  -  lettered   edges   of   the  base   abcde;    join    the   extremities   of   these   chords  with 
V:    then   on   D  V  lay   off  DP=d'p';    make    C0=  EQ=  d'  o'"  =  the  real   length   of   c'o';     also   B  X= 
A  M=  d'm'"  =  the    actual    length    of    a'm'  and    b'n';    join    the    points    /',  0,   etc.,   thus    obtaining  the 
development   of  the   outer  boundary   of  the   section.      The   pattern   A1B1CDE1   of  the   base  is   obtained 
from   the    plan  in   Fig.  238,   while   XM/j.2p.,o.,   is   a   duplicate   of  the  shaded   part    of   the    xectioiial    view 
in   the   same   figure. 

(7)  In  making  a   model    of   the   pyramid   the   student   should   use  heavy    Bristol    board,    and    make 
allowance,   wherever  needed,   of  an   extra   width    for  overlap,   slit   as   at   .c,   y  and   z  (Fig.  239).      On   this 


r.  ESS- 


overlap  put  the   mucilage   which   is   to   hold   the   model    in    shape.      The    faces    will    fold    better    if   the 
Bristol   board   is   cut   half  way   through   on   the   folding   edge. 

397.  For  convenient   reference   the   characteristic   features   of  the   Third   Angle   Method,   all    of  which 
have   now   been   fully   illustrated,   may   thus   be   briefly   summarized : 

(a)  The    various    views    of   the    object   are   so  grouped    that  the    plan   or   top  view  comes   abore  the 
front   elevation;    that   of  the   bottom   below  it;     and    analogously    for    the    projections    of   the    right    and 
left  sides. 

(b)  Central,  reference   planes   are   taken  through   the   various   views,  and,  in   each   view,  the  distance 
of   any    point    from    the    trace    of   the   central   plane   of   that    view   is   obtained   by   direct   transfer,   with 
the    dividers,   of   the    distance    between    the    same    point    and    reference    plane,   as    seen    in    some    other 
view,   usually   the   plan. 

398.  To   draw  a  truncated,    pyramidal  block,   having    a    rectangular    retex*    in    its  top;     angle    of    sides, 
60°;    lower  base   a   rectangle    3"  x  2",   having  its   longer  sides   at   30°   to    the    horizontal;     total    height 
&";    recess   1^"  X  ft",  and   \"   deep.      (Fig.  240.) 

The   small   oblique   projection   on   the   right   of  the   plan   shows,   pictorially,   the   figure  to   be   drawn. 


PROJECTION    OF   SOLIDS.— WORKING    DRAWINGS. 


141 


The  plan  of  the  lower  base  will  be  the  rectangle  abdc,  ?>"  x  2",  whose  longer  edges  are  inclined 
30°  to  the  horizontal. 

Take  A  B  and  mn  as  the  H- traces  of  auxiliary,  vertical  planes,  perpendicular  to  the  side  and 
end  faces  of  the  block.  Then  the  sloping  face  whose  lower  edge  is  d  e,  and  which  is  inclined  60  ° 
to  H,  will  have  dty  for  its  trace  on  plane  m  u.  A  parallel  to  mn  and  ^"  from  it  will  give  x,,  the 
auxiliary  projection  of  the  upper  edge  of  the  face  vved,  whence  sv — at  first  indefinite  in  length — is 
derived,  parallel  to  de.  Similarly  the  end  face  btxd  is  obtained  by  projecting  db  upon  A  B  at  ft,, 
drawing  ft,z  at  (50°  to  A  B  and  terminating  it  at  *,  by  CD,  drawn  at  the  same  height  (-£$")  as 
before.  A  parallel  to  bd  through  *2  intersects  ex,  at  *,  giving  one  corner  of  the  plan  of  the  upper 
base,  from  which  the  rectangle  *  t  u  v  is  completed,  with  sides  parallel  to  those  of  the  lower  base. 


As  the  recess  has  vertical  sides  we  may  draw  its  plan,  o  p  q  r,  directly  from  the  given  dimen- 
sions, and  show  the  depth  by  short -dash  lines  in  each  of  the  elevations. 

The  ordinary  elevations  are  derived  from  the  plan  as  in  preceding  problems;  that  is,  for  the 
front  elevation,  a'u's'd',  by  verticals  through  the  plans,  terminating  according  to  their  height,  either 
on  n'd'  or  on  «'«',  fi,"  above  it.  For  the  side  elevation,  e"v"t"b",  with  the  heights  as  in  the  front 
elevation,  the  distances  to  the  right  or  left  of  x"  equal  those  of  the  plans  of  the  same  points  from 
« /,  regarding  the  latter  as  the  h.  t,  of  a  central,  vertical  plane,  parallel  to  V. 

The  plane  ST  of  right  .-section,  perpendicular  to  the  aria  KL,  cuts  the  block  in  a  section  whose 
true  size  is  shown  in  the  line -tinted  figure  r/,  /i,  k,  /,,  and  whose  construction  hardly  needs  detailed 
treatment  after  what  has  preceded.  The  shaded,  longitudinal  section,  on  central,  vertical  plane  K L, 
also  interprets  itself  by  means  of  the  lettering. 


142 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


The  true  size  of  any  face,  as  a  u  v  e,  may  be  shown  by  rabatment  about  a  horizontal  edge,  as  a  e. 
As  v  is  actually  -jY'  above  the  level  of  e,  we  see  that  ve  (in  space)  is  the  hypothenuse  of  a  triangle 
of  base  ve  and  altitude  -fa".  Construct  such  a  triangle,  v  v,e,  and  with  its  hypothenuse  «2e  as  a 
radius,  and  e  as  a  centre,  obtain  i>,  on  a  perpendicular  to  ae  through  v  and  representing  the  path 
of  rotation.  Finding  ul  similarly  we  have  aulvle  as  the  actual  size  of  the  face  in  question. 

If  more  views  were  needed  than  are  shown  the  student  ought  to  have  no  difficulty  in  their 
construction,  as  no  new  principles  would  be  involved. 

399.  To  draw  a  hollow,  pentagonal  prism,  2"  long;  edges  to  be  horizontal  and  inclined  35°  to 
V ;  base,  a  regular  pentagon  of  1 "  sides ;  one  face  of  the  prism  to  be  inclined  60  °  to  H ;  distance 
between  inner  and  outer  faces, 


In  Fig.  241  let  HK  be  parallel  to  the  plans  of  the  axis  and  edges;  it  will  make  35°  with  a 
horizontal  line.  Perpendicular  to  H  K  draw  m  n  as  the  h.  t.  of  an  auxiliary,  vertical  plane,  upon 
which  we  may  suppose  the  base  of  the  prism  projected.  In  end  view  all  the  faces  of  the  prism 
would  be  seen  as  lines,  and  all  the  edges  as  points.  Draw  a1bl,  one  inch  long  and  at  60°  to  mn, 
to  represent  the  face  whose  inclination  is  assigned.  Completing  the  inner  and  outer  pentagons, 
allowing  \"  for  the  distance  between  faces,  we  have  the  end  view  complete.  The  plan  is  then 


PROJECTION    OF    SOLIDS.  —  WORKING    DRAWINGS.  143 

t 
obtained   by   drawing   parallels  to   HK  through   all  the   vertices    of   the    end    view,   and  terminating  all 

by   vertical   planes,   a  d  and  g  h ,   parallel  to  m  n  and   2 "  apart. 

The  elevations  will  be  included  between  horizontal  lines  whose  distance  apart  is  the  extreme  height 
z  of  the  end  view;  and  all  points  of  the  front  elevation  are  on  verticals  through  their  plans,  and  at 
heights  derived  from  the  end  view.  The  most  expeditious  method  of  working  is  to  draw  a  horizontal 
reference  line,  like  that  of  Fig.  243,  which  shall  contain  the  lowest  edge  of  each  elevation;  measuring 
upward  from  this  line  lay  off,  on  some  random,  vertical  line,  the  distance  of  each  point  of  the  end 
view  from  a  line  (as  the  parallel  to  mn  through  bl  in  Fig.  241,  or  xy  in  Fig.  243)  which  repre- 
sents the  intersection  of  the  plane  of  the  end  view  by  a  horizontal  plane  containing  the  lowest  point 
or  edge  of  the  object;  horizontal  lines,  through  the  points  of  division  thus  obtained,  will  contain  the 
projections  of  the  corners  of  the  front  elevation,  which  may  then  be  definitely  located  by  vertical 
lines  let  fall  from  the  plans  of  the  same  points.  For  example,  e'  and  /',  Fig.  241,  are  at  a  height, 
z,  above  the  lowest  line  of  the  elevation,  equal  to  the  distance  of  el  from  the  dotted  line  through 
bl;  or,  referring  to  Fig.  243,  which,  owing  to  its  greater  complexity,  has  its  construction  given  more 
in  detail,  the  distance  upward  from  M  to  line  G  is  equal  to  #,</.,  on  the  end  view;  from  M  to  Q 
equals  qlq2,  and  similarly  for  the  rest. 

Since  the  profile  plane  is  omitted  in  Fig.  241  we  take  M'  N'  to  represent  the  trace  upon  it  of 
the  auxiliary,  central,  vertical  plane  whose  h.  t.  is  MN;  as  already  explained,  all  points  of  the  side 
elevation  are  then  at  the  same  level  as  in  the  front  elevation,  and  at  distances  to  the  right  or  left 
of  M' N'  equal  to  the  perpendicular  distances  of  their  plans  from  MN.  For  example,  e"s"  equals  e  s. 

The  shade  lines  are  located  on  the  end  view  on  the  assumption  that  the  observer  is  looking 
toward  it  in  the  direction  HK. 

400.  Projections  of  a  hollow,  pentagonal  prism,   cut  by   a  vertical  plane  oblique  to  V.      Letting  the  data 
for  the   prism   be  the  same   as  in  the   last  problem,   we   are  to   find    what    modification  in  the   appear- 
ance of  the  elevations  would  result  from  cutting  through  the  object  by  a  vertical  plane  PQ  (Fig.  242) 
and  removing  the   part  hxdi  which  lies  in  front  of  the  plane  of  section. 

Each  vertex  of  the  section  is  on  an  edge  of  the  elevation  and  is  vertically  below  the  point  where 
P  Q  cuts  the  plan  of  the  same  edge ;  the  student  can,  therefore,  readily  convert  the  elevations  of 
Fig.  241  into  reproductions  of  those  of  Fig.  242  by  drawing  across  the  plan  of  Fig.  241  a  trace  P  Q, 
similarly  situated  to  the  P  Q  of  Fig.  242.  Supposing  that  done,  refer  in  what  follows  to  both 
Figures  241  and  242. 

Since  P  Q  contains  h  we  find  h'  as  one  corner  of  the  section.  Both  ends  of  the  prism  being 
vertical,  they  will  be  cut  by  the  vertical  plane  PQ  in  vertical  lines;  therefore  h'l'  is  vertical  until 
the  top  of  the  prism  is  reached,  at  I'.  Join  V  with  x',  the  latter  on  the  vertical  through  x — the 
intersection  of  PQ  with  the  right-hand  top  edge  ed,  e'd'.  From  x  the  cut  is  vertical  until  the 
interior  of  the  prism  is  reached,  at  o',  on  the  line  5-4.  We  next  reach  iv'  on  edge  No.  4.  The 
line  o'w'  has  to  be  parallel  to  x'l'  (two  parallel  planes  are  cut  by  a  third  in  parallel  lines);  but 
from  w'  the  interior  edge  of  the  section  is  not  parallel  to  I'h',  since  PQ  is  not  cutting  a  vertical 
end,  but  the  inclined,  interior  surface.  The  other  points  hardly  need  detailed  description,  being 
similarly  found. 

The  side  elevation  is  obtained  in  accordance  with  the  principle  fully  described  in  Art.  396  and 
summarized  in  Art.  397  (b).  M' N'  represents  the  same  plane  as  MN;  e"s"  equals  es,  and  anal- 
ogously for  other  points. 

401.  In    his    elementary    work    in    projections    and    sections   of  solids   the  student  is   recommended 
to    lay    an    even    tint    of   burnt    sienna,   medium    tone,   over    the    projections   of  the   object,   after  which 


144 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


any  section  may  be  line-tinted;  and,  if  he  desires  to  further  improve  the  appearance  of  the  views, 
distinctions  may  be  made  between  the  tones  of  the  various  surfaces  by  overlaying  the  burnt  sienna 
with  flat  or  graded  washes  of  India  ink. 


SIDE  ELEVATION 

IN' 

402.  Projections  of  an  L- shaped  block,  after  being  cut  by  a  plane  oblique  to  both  V  and  H;  the  block 
ako  to  be  inclined  to  V  and  H,  and  to  have  running  through  it  two,  non-communicating,  rectangular  openings, 
whose  directions  are  mutually  perpendicular. 

If  the  dotted  lines  are  taken  into  account  the  front  elevation  in  Fig.  243  gives  a  clear  idea  of 
the  shape  of  the  original  solid.  The  end  view  and  plan  give  the  dimensions. 

Requiring  the  horizontal  edges  of  the  block  to  be  inclined  30°  to  V,  draw  the  first  line  xy  at 
60°  to  the  horizontal;  the  plans  of  all  the  horizontal  edges  will  be  perpendicular  to  xy. 

Let  the  inclination  of  the  bottom  of  the  block  to  H  be  20°.  This  is  shown  in  the  end  view 
by  drawing  mlpl  at  20°  to  xy.  All  the  edges  of  the  end  view  of  the  object  will  then  be  parallel 
or  perpendicular  to  mlpl  and  should  be  next  drawn  to  the  given  dimensions. 


WORKING     DRAWINGS.  — PLANE    SECTIONS     OF    SOLIDS. 


145 


PLAN 


The  central  opening,  b,dt «,<>,,  through  the  larger  part  of  the  block,  has  its  faces  all  ^"  from 
the  outer  faces.  In  the  plan  this  is  shown  by  drawing  the  lines  lettered  of  at  a  distance  of  \" 
from  the  boundary  lines,  which  last  are  indicated  as  H"  apart. 

The  opening  q^s,*, 
has  three  of  its  faces  i" 
from  the  outer  surfaces 
of  the  block,  while  the 
fourth,  <?,»-,,  is  in  the 
same  plane  as  the  outer 
face  h ,  e , . 

The  cutting  plane 
A""  F  gives  a  section 
which  is  seen  in  end 
view  in  the  lines  e^,, 
i1jl  and  £,?,;  while  in 
plan  the  section  is  pro- 
\  :K-  jected  in  the  shaded 

'•              \  portion,     obtained,     like 

\  all    other    parts    of    the 

\  \  plan,   by    perpendiculars 

\       -*—  to     xy     from     all     the 

.—  :'M'\" 

points  of  the  end  view. 
For  the  front  eleva- 
tion draw  first  the  "reference  line."  To 
provide  against  overlapping  of  projections 
the  reference  line  should  be  at  a  greater 
distance  below  the  lowest  point,  /,  of  the 
plan,  than  the  greatest  height  («,«2)  of 
the  end  view  above  x  y.  Then  on  M  W 
lay  off  from  M  the  heights  of  the  vari- 
ous horizontal  edges  of  the  block,  deriving 
them  from  the  end  view.  Thus  a,a.2  is 
the  height  of  An'  from  M;  from  M  to 
level  B  equals  bj>.n  etc.  Next  project  to 
the  level  A  from  points  a  a  of  the  plan, 
getting  edge  a' a'  of  the  elevation,  and 
similarly  for  all  the  other  corners  of  the 
block.  Notice  that  all  lines  that  are  parallel 
on  the  object  will  be  parallel  in  each  projec- 
tion (except  when  their  projections  coin- 
cide); also  that  in  the  case  of  sections,  those 
outlines  will  be  parallel  which  are  the  inter- 
section of  parallel  planes  by  a  third  plane. 

These   principles   may  be   advantageously   employed    as   checks   on   the   accuracy    of  the   construction  by 

points.      The   construction   of  the   side   elevation   is   left  to   the   student. 


146 


THEORETICAL     AND    PRACTICAL     GRAPHICS. 


E— 


FRONT  ELEVATION. 
With  section  made  by  vertical  plane  P  Q 
Reference  line 


SIDE   ELEVATION. 

With  section  by  plane  S  T.  Shade 
lines  on  this  view  are  located  for 
pictorial  effect  and  not  in  accord- 
ance with  shop  rule. 


WORKING    DRAWINGS.  — PLANE    SECTIONS    OF    SOLIDS.  147 

403.  Projections    and    sections    of   a    block    of  irregular  form,  with    two    mutually   perpendicular  openings 
through  it,  and  with  equal,  square  frames   projecting  from  each  side. 

In  Fig.  244  the  side  elevation  shows  clearly  the  object  dealt  with,  while  we  look  to  the  end 
view  for  most  of  the  dimensions.  The  large  central  opening  extends  from  w,ws  to  xlyl.  The  width  of 
the  main  portion  of  the  block  is  shown  in  plan  as  2^",  between  the  lines  lettered  ae.  The  square 
frames  project  |"  from  the  sides,  while  the  width  of  the  central  opening  between  the  lines  wx  is  f". 

Two  section  planes  are  indicated,  S  T  across  the  end  view,  and  P  Q — a  vertical  plane — across  the 
plan ;  the  section  made  by  plane  S  T  is,  however,  shown  only  in  fringed  outline  on  the  plan,  though 
fully  represented  on  the  side  elevation.  The  front  elevation  shows  the  section  made  by  plane  P  Q, 
with  the  visible  portion  of  that  part  of  the  object  that  is  behind  the  cutting  plane. 

Although  detailed  explanation  of  this  problem  is  unnecessary  after  what  has  preceded,  yet  a 
brief  recapitulation  of  the  various  steps  in  the  construction  of  the  views  may  be  appreciated  by  some, 
before  passing  on  to  a  more  advanced  topic. 

(a)  E  F,  the    first    line    to    draw,    is    the    trace    of  the    vertical    plane    on    which  the  end  view  is 
projected,  and  is   at   an    angle    of   60  °    to  a  horizontal  line  in  order  that  the  edges  of  the  object  (as 

a  a,  bb ee)    may    be    inclined    at    30  °    to    the    front  vertical  plane,  which  we  may  assume  as  one 

of  the   conditions   of  the   problem. 

(b)  A   rotation   of   the   object  through   an   angle   6°   about    a    horizontal    axis    that  is   perpendicular 
to  E  F,  as,  for    example,  the  edge  through  /,  is  shown   by   the   inclination    of   the    end  view    to  E  F 
at  an  angle  al/lE=0°. 

(c)  Drawing  the  end  view  at  the  required  angle    to  EF  we    next    derive    the  plan  therefrom  by 
perpendiculars  to   E  F,  terminating    them  on  parallels  to  EF   (as    the    lines    ae,   wx,   nh,   etc.,)  whose 
distances   apart   conform    to   given   data. 

(d)  The  elevations.     For  these  a  common  reference    line    E'  F'  f"  is  taken,  horizontal,  and  sufficiently 
below   the   plan  to   avoid  an   overlapping  of  views. 

For  the  front  elevation  any  point  as  b',  is  found  vertically  below  its  plan  6,  and  is  as  far  from 
E'  F'  as  bl  is — perpendicularly —  from  E  F. 

The  height  at  which  the  section  plane  P  Q  cuts  any  line  is  similarly  obtained.  Thus  at  z  it 
cuts  the  vertical  end  face  of  the  block  in  a  line  which  is  carried  over  on  the  end  view  in  the 
indefinite  line  Zz2;  the  portions  of  Zz2  which  lie  on  the  end  view  of  the  frame  gihlil  are  the 
only  real  parts  to  transfer  to  the  front  elevation,  and  are  seen  on  the  latter,  vertically  below  z  and 
running  from  z'  down;  their  distances  from  E' F'  being  simply  those  from  Z,  on  the  end  view, 
transferred. 

The  side  elevation.  Any  point  or  edge  is  at  the  same  level  on  the  side  elevation  as  on  the 
front;  hence  the  edge  through  b"  is  on  b' b'  produced.  The  distances  to  the  right  or  left  of  M' N 
equal  those  of  the  corresponding  points  on  the  plan  from  M  N;  thus  o"j'  equals  oj,  etc. 

404.  Changed   planes    of  projections.       In    the    problems    of   Arts.   399-403    the    employment    of   an 
"end   view"  —  which    was   simply   an   auxiliary  elevation — has   prepared    the    student    for  the  further  use 
of  planes   other  than   the   usual   planes   of  projection;     and  if   the  auxiliary   plan    is    now    mastered    he 
is    prepared    to    deal    with  any  case  of   rotation  of  object  about  vertical   or  horizontal  axes,  since  new 
and  properly  located  planes   of  projection  are  their  practical  equivalent. 

In  Fig.  245  the  object  is  represented  in  its  initial  position  by  the  line-tinted  figures  marked 
"first  plan"  and  "first  elevation."  The  third  and  fourth  elevations  show  somewhat  more  pictorially 
that  it  is  a  hollow,  truncated,  triangular  prism,  having  through  it  a  rectangular  opening  that  is  per- 
pendicular to  the  front  and  rear  faces. 


148 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


reference  line  for  fourth  elevation.     \ 
" " 


R, 


CHANGING    PROJECTION-PLANE    EQUIVALENT    TO    ROTATING     OBJECT.      149 

(a)  Rotation    about    a    vertical    axis,   or    its    equivalent,   a    change    in    the    vertical   plane    of  projection. 

Kesult :    second  elevation  derived   from  first   plan  and  elevation. 

Let  the  axis  be  one  of  the  vertical  edges  of  the  object,  as  that  at  d  in  the  first  plan ;  also  let 
the  rotation  be  through  an  angle  Ydo  or  6°,  (6  being  taken,  for  convenience,  equal  to  the  angle 
YaF,  which  —  with  the  line  pq — will  be  employed  in  a  later  construction).  If  we  were  actually  to 
rotate  the  object  through  an  angle  6  the  new  plan  would  be  the  exact  counterpart  of  the  first,  but 
its  horizontal  edges  would  make  an  angle  6  with  their  former  direction,  and  the  new  elevation 
would  partly  overlap  the  first  one.  To  avoid  the  latter  unnecessary  complication,  as  also  the 
duplication  of  the  plan,  we  make  the  first  plan  do  double  duty,  since  we  can  accomplish  the 
equivalent  of  rotation  of  the  object  by  taking  a  new  vertical  plane  that  makes  an  angle  0  with  the 
plane  on  which  the  first  elevation  was  made.  This  equivalence  will  be  more  evident  if  some  small 
object,  as  a  piece  of  india  rubber,  is  placed  on  the  "  first  plan "  with  its  longer  edges  parallel  to 
a  b,  and  is  then  viewed  in  the  direction  of  arrow  No.  2  through  a  pane  of  glass  standing  vertically 
on  XZ;  after  which  turn  both  the  object  and  the  glass  through  the  angle  6  until  the  glass  stands 
vertically  on  e'j'  and  then  view  in  the  direction  of  arrow  No.  1. 

The  second  plane  may  be  located  anywhere,  as  long  as  the  angle  6  is  preserved;  XZ,  making 
angle  &  at  xl  with  e'j',  is,  therefore,  a  random  position  of  the  new  plane,  and  the  projection  upon 
it  is  our  "second  elevation." 

Since  the  heights  of  the  various  corners  of  an  object  remain  unchanged  during  rotation  about  a 
vertical  axis  we  will  find  all  points  of  the  second  elevation  at  distances  from  the  reference  line  XZ 
that  are  derived  from  the  first  elevation,  and  laid  off  on  lines  drawn  perpendicular  to  XZ  from  the 
vertices  of  the  plan:  thus  a  0  is  perpendicular  to  XZ,  and  0 a"  equals  o'a';  c"J'  equals  e'j',  etc. 

(b)  Rotation    about  a    horizontal    axis,    or    its    equivalent,    the    adoption    of    a    new    horizontal    plane. 

Result :     second   plan  derived   from  first  plan  and  second  elevation. 

Having  in  the  last  case  illustrated  the  method  of  complying  with  the  condition  that  rotation 
should  occur  through  a  given  angle  (which  is  incidentally  shown  again,  however,  in  the  next  con- 
struction) we  now  choose  an  axis  p  q  so  as  to  illustrate  a  different  kind  of  requirement,  viz. :  that 
during  rotation  the  heights  of  any  two  points  of  the  object,  which  were  at  first  at  the  same  level  as 
the  axis,  shall  be  in  some  predetermined  ratio,  regardless  of  the  amount  of  rotation.  In  the  figure 
it  is  assumed  that  e'(rf)  is  to  be  at  one-fifth  the  height  of  j'j,  and  that  rotation  shall  occur  about 
an  axis  passing  through  the  lower  end  o'  of  the  vertical  edge  at  a.  By  drawing  ad  and  aj, 
dividing  the  latter  into  five  equal  parts,  and  joining  d  with  n  —  the  first  point  of  division  from  a — - 
we  obtain  the  direction  dn,  parallel  to  which  the  axis  p  q  is  drawn  through  o.  The  distance  dp  is 
then  one -fifth  of  jq,  and  they  shorten  in  the  same  ratio,  as  rotation  occurs. 

After  locating  the  axis  the  next  step  is,  invariably,  the  drawing  of  an  elevation  upon  a  plane 
perpendicular  to  the  axis.  This  we  happen,  however,  to  have  already  in  our  "  second  elevation," 
having,  in  the  interest  of  compactness,  so  taken  6  in  the  preceding  case  that  the  vertical  plane  XZ 
would  be  perpendicular  to  the  axis  we  are  now  ready  to  use. 

Any  rotation  of  the  object  about  p  q  will,  evidently,  not  change  the  form  of  the  "  second 
elevation"  but  simply  incline  it  to  XZ.  But,  as  before,  instead  of  actually  rotating  the  object, 
which  would  probably  give  projections  overlapping  those  from  which  we  are  working,  we  adopt  a 
new  plane  MN  as  a  horizontal  plane  of  projection,  so  taken  that  it  fulfills  either  of  the  following 
conditions:  (a)  that  the  object  should  be  rotated  about  p  q  through  an  angle  J'  0  N  =  ft;  (b) 
that  the  corner  J'  should  be  higher  f"  n  0  by  an  amount  x,  MN  being  drawn  tangent  to  an 
arc  having  J'  for  its  centre,  and  J'f  to  x)  for  its  radius. 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 

Reference  to  Fig.  246  may  make  it  clearer  to  some  that  AT  TV  is  the  trace  of  the  new  plane 
upon  the  vertical  plane  whose  h.  t.  is  XZ;  that  ON  lies  vertically  below  the  line  XZ  and  is  as  truly 
perpendicular  to  the  axis  of  rotation  as  is  XZ;  also  that  in  Fig.  245  a  view  in  the  direction  of 
arrow  No.  3  (i.  e.,  perpendicular  to  M N)  is  equivalent  to  a  view 
perpendicular  to  the  plane  V  in  Fig.  246  after  the  whole  assem- 
blage of  planes  and  object  has  been  rotated  together  about 
HOH  until  the  "new  plane"  takes  the  position  out  of 
which  the  first  horizontal  plane  has  just  been  rotated. 

(The  remainder  of  the   references  are  to  Fig.  245.) 

The  second   plan  is  obtained   by    drawing  Pl  Q17 
parallel     to    MN,     to    represent    the     transferred 
trace   PQ    of   a    vertical   reference   plane  taken 
through    some   edge   b  and    parallel   to    the 
plane    Z  0  N   of   the    second    elevation ; 
then    any    point    dl    is    as    far    from 
Pl  Q ,    as    the    same    point    d    on 
the   first    plan    is    from    P  Q ; 
similarly,    from    point    wl  to 
Pj  Q,  equals   distance  wb. 

(c)  Further  rotation  about  ver- 
tical axes.  To  show  how  the  foregoing  processes  may 
be  duplicated  to  any  desired  extent  let  us  suppose 
that  the  object,  as  represented  by  the  second  plan 
and  elevation,  is  to  be  rotated  through  an  angle  <f> 
about  a  vertical  line  through  &,.  If  the  rotation 
actually  occurred,  the  plan  b1Gl  would  take  the  posi- 
tion blG^,  and  the  other  lines  of  the  plan  would  take  corresponding  positions  in  relation  to  a  vertical 
plane  on  PjQ,.  A  new  vertical  plane  on  blQ3,  at  an  angle  <£  to  blG1,  will,  however,  evidently 
hold  the  same  relation  to  the  plan  as  it  stands,  and  transferring  such  new  plane  forward  to  0'  Rj 
we  then  obtain  the  points  of  the  new  (third)  elevation  by  letting  fall  perpendiculars  to  0'  Rl  from 
the  vertices  of  the  second  plan,  and  on  them  laying  off  heights  above  0' Rl  equal  to  those  of  the 
same  points  above  MN  in  the  second  elevation.  Thus  j'  9  equals  J'f;  W  6  in  the  fourth  equals 
W"G  in  the  second. 

The  fourth  elevation  is  a  view  in  the  direction  of  arrow  No.  5,  giving  the  equivalent  of  a  ninety - 
degree  rotation  of  the  object  from  its  last  position.  To  obtain  it  take  a  reference  line  r  r  through 
some  point  of  the  second  plan,  and  parallel  to  0'  Rl;  then  R  R'  represents  the  vertical  plane  on  r  r, 
transferred.  From  R  R'  lay  off — on  the  levels  of  the  same  points  in  the  third  elevation — distance 
lC"'=c,C1j  4W"  =  w2wI)  as  in  preceding  analogous  constructions. 

THE    DEVELOPMENT    OF    SURFACES. 

405.  The  development  of  surfaces  is  a  topic  not  altogether  new  to  the  student  who  has  read 
Chapter  V  and  the  earlier  articles  of  this  chapter;*  so  far,  however,  it  has  occurred  only  incidentally, 
but  its  importance  necessitates  the  following  more  formal  treatment,  which  naturally  precedes  problems 
on  the  interpenetration  of  surfaces,  of  which  a  "  development "  is  usually  the  practical  outcome. 


*The  following  articles  should  be  carefully  reviewed  at  this  point:    120;  191;  344-6;  389,  and  Case  6  of  Art.   396. 


THE  DEVELOPMENT  OF  SURFACES. 


151 


A  development  of  a  surface,  using  the  term  in  a  practical  sense,  is  a  piece  of  cardboard  or,  more 
generally,  of  sheet -metal,  of  such  shape  that  it  can  be  either  directly  rolled  up  or  folded  into  a 
model  of  the  surface.  Mathematically,  it  would  be  the  contact- area,  were  the  Surface  rolled  out  or 
unfolded  upon  a  plane. 

The  "shop"  terms  for  a  developed  surface  are  "surface  in  the  flat,"  "stretch-out,"  "roll -out"; 
also,  among  sheet -metal  workers  it  is  called  a  pattern;  but  as  pattern -making  is  so  generally  under- 
stood to  relate  to  the  patterns  for  castings  in  a  foundry,  it  is  best  to  employ  the  qualifying  words 
sheet- metal  when  desiring  to  avoid  any  possible  ambiguity. 

406.  The  mathematical  nature  of   the  surfaces  that  are  capable  of    development  has  been  already 
discussed   in   Arts.   344-346.      Those    most    frequently    occurring    in    engineering  and    architectural   work 
are  the   right  and   oblique   forms   of  the   pyramid,   prism,   cone   and   cylinder. 

407.  In   Art.  120   the  development  of  a  right  cylinder   is   shown    to    be    a    rectangle    of    base    equal  to 
2irr  and   altitude   h,   where   h  is   the   height   of  the   cylinder  and   r  is   the   radius   of  its   base. 

408.  The    development    of   a    right    cone    is    proved,   by    Art.  191,  to    be    a    circular    sector,    of   radius 
equal    to    the    slant    height    R    of   the   cone,   and   whose   angle   6  is   found   by   means   of  the    proportion 
R  :  r  ::  360°  :  0;    r  being  the  radius   of  the  base  of  the  cone. 

409.  The  development  of  a  right  pyramid  is   illustrated   in   Art.  389,   and   in   Case   6  .of  Art.  396. 

410.  We    next    take    up    right    and    oblique   prisms,    and    the    oblique   pyramid,    cone    and    cylinder; 
while   for  the  sake   of    completeness,   and   departing    in    some   degree    from    what    was    the  plan    of   this 
work   when   Arts.  345   and   346   were   written,   the  regular  solids    will   receive   further  treatment,   and   also 
the   developable  helicoid. 

411.  The  development  of  a  right  prism.  Fig. 
247  represents  a  regular,  hexagonal  prism.  The 
six  faces  being  equal,  and  e  b  c  f  showing  their 
actual  size,  we  make  the  rectangles  A  B  G  D , 
BEFC,  etc.,  each  equal  to  ebcf;  then  AAl 
equals  the  perimeter  of  the  upper  base,  and  we 


ngr.  E-i'7- 

a         A           B           E           H                                    A! 

<! 

B, 

Taking  the  same  prism  as  in  the 


have    the    rectangle    A  A ,  B ,  D   for  the  development  sought. 

412.  The    development    of   a    right   prism    below    a   cutting   plane. 
last    article    develop    first    as    if    there    were     no 

section  to  be  taken  into  account.  This  gives,  as 
before,  a  rectangle  of  length  A  A  l  and  of  altitude 
ad,  divided  into  six  equal  parts.  Then  project, 
from  each  point  where  the  plane  cuts  an  edge, 
to  the  same  edge  as  seen  on  the  development. 

413.  Right   section.      Rectified    curve.      Developed   curve.      A    plane    perpendicular    to    the    axis    of    a 
surface    cuts    the    latter    in    a    right  section.      The  bases   of  right   cones,   pyramids,   cylinders   and   prisms 
fulfil    this    condition   and   require   no   special   construction  for  their  determination;    but  the  development 
of  an    oblique  form  usually   involves  the   construction   of   a    right  section    and    then  the  laying   off   on 
a  straight    line    of   a    length  equal  to  the   perimeter    of   such    section.      Should  the  right  section  be  a 
curve  its   equivalent  length   on   a  straight  line  is  called  its  rectification,  which   should   not  be  confounded 
with   its   development,  the  latter   not  being   necessarily   straight. 

414.  The    development    of   an    oblique   prism,    when    the  faces    are    equal    in    width.      In    Fig.   249    an 
oblique,  hexagonal  prism    is    shown,    with    x    for    the    width    of   its    faces.      Since    the    perimeter    of  a 
right    section    would    evidently    equal  6z  we  may  directly  lay  off   x  six  times  on  some  perpendicular 


152 


THEORETICAL    AXD    PRACTICAL    GRAPHICS. 


FLg.  SSO. 


to  the  edges,  as  that  through  a.  The  seven  parallels  to  ab,  drawn  at  distances  x  apart,  will  contain 
the  various  edges  of  the  prism  as  it  is  rolled  out  on  the  plane;  and  the  positions  of  the  extremities 
are  found  by  perpendiculars  from  their  original  positions. 
The  initial  position  a ,  b ,  is  parallel  to  but  at  any  dis- 
tance from  ab.  The  base  edges  are  evidently  unequal. 
415.  The  development  of  an  oblique  prism  whose  faces 
are  unequal  in  width. 

In  Fig.  250  c' d' h' g'  is  the  elevation  of  the  prism; 
np  a  plane  of  right  section.  To  get  1-2-3-4,  the 
true  shape  of  the  right  section,  we  require  a  b  h  f  e,  the 
plan  of  the  prism.*  Assuming  that  to  have  been  given 
imagine  next  a  vertical  reference  plane  standing  on  a  6. 
The  right  section  plane  np  cuts  the  edge  c' d'  at  n, 
which  is  at  a  distance  x  in  front  of  the  assumed  reference  plane.  Make  n  2  =  x.  Similarly  make 

1  =  2/,   and  j?4  =  z;    then   1-2-3-4   is    the    right    section,   seen 
in   its   true   size   after   being    revolved    about    the    trace    of   the 
right -section   plane   upon  the  assumed   reference  plane. 

Prolong  p  n   indefinitely,   and   on   its   extension  make 
l'-2'=l-2;  2'-3'  =  2-3,  etc.    Parallels  to  c' d'  through 
the  points  of  division  thus  obtained  will  contain  the 
edges   of  the   developed    prism,  and    their    lengths 
are  definitely   determined    by   perpendiculars,   as 
\     /     \  h' h",  f'f",  from   the  extremities   of  the  orig- 

inal edges. 

416.  The  development  of  an  oblique 
cylinder,  having  a  circular  base  and  elliptical 
right  section.  Let  am'n'k,  Fig.  251,  be  an 
oblique  cylinder  with  circular  base.  Take 
any  plane  of  right  section,  as  a'k'. 
Draw  various  elements,  as  those  through 
b',  c',  etc.,  and  from  their  lower  extrem- 
ities erect  perpendiculars  to  ak,  as  cc,, 
terminating  them  on  the  arc  aftk,  which 
represents  the  half  base  of  the  cylinder. 
On  cc'  make  c'c"  =  cc1;  on  e  e'  take 
e'e"  =  ee1,  and  similarly  obtain  other 
points  on  the  elements,  through  which  the  curve  a' c"  e"  g"  k'  can  be  drawn,  this  being  one -half  of 
the  curve  of  right  section,  shown  after  revolution  about  its  shorter  diameter.  Making  KA  equal  to 
the  rectified  semi-ellipse  just  obtained,  lay  off  A  C  =  arc  a'c";  C.E=arc  c"e",  etc.,  and  through  the 
points  of  division  thus  obtained  on  KA  draw  indefinite  parallels  to  the  axis  of  the  cylinder.  These 
will  represent  the  elements  on  the  development,  and  are  limited  by  the  dotted  lines  drawn  per- 
pendicular to  the  original  elements  and  through  their  extremities. 

The  area  a1k1NM  is  the  development  of   one -half  of  the  cylinder,  the  shaded  area  representing 
all  between  a'k'  and  the  base  ak. 

•In  the  interest  of  compactness  the  "First  Angle"  position  of  the  views  (Art.  385)  is  employed  in  Figs.  250,  253  and  255. 


THE    DE  VE  LOPMENT    0  F    ft  UR  FACE  S. 


153 


417. 


vc    to 
length 


The  development  of  an  oblique  pyramid.     The   development   will   evidently   consist   of  a  series  of 

triangles  having  a  common  vertex.  To  ascertain 
the  length  of  any  edge  we  may  carry  it  into  or 
parallel  to  a  plane  of  projection.  Thus  in  Fig.  252  the 
edge  vb  is  carried  into  the  vertical  plane  at  vb".  Its 
true  length  is  the  hypothenuse  of  a  right-angled 
triangle  of  base  ob  =  ob",  and  altitude  v  o . 

In  Fig.  253  a  pyramid  is  shown  in  plan  and 
elevation.  Making 
o  a"  =  v  a  we  have 
v' a"  for  the  actual 
length  of  edge  v' a',  a 
construction  in  strict 
>M  analogy  to  that  of 
Fig.  252.  The  plan 
v  b  being  parallel  to 
the  base  line  shows 
that  v'1>'  is  the 
actual  length  of  that 
edge.  By  carrying 
vct,  where  it  becomes  parallel  to  V,  and  then  projecting  cl  to  c"  we  get  v' c"  for  the  true 

of  edge  v'c'.  FIS.  253. 

A, 


To  illustrate  another  method   make   v  v ,  =  o '  o ;    then   v ,  d 
by  rabatment  into   H. 


u,        is  the  real  length  of  v'd',  shown 


154 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


For  the  development  take  some  point  vt  and  from  it  as  a  centre  draw  arcs  having  for  radii  the 
ascertained  lengths  of  the  edges.  Thus,  letting  v2A  represent  the  initial  edge  of  the  development, 
take  A  as  a  centre,  ad  as  a  radius,  and  cut  the  arc  of  radius  vtd  at  D;  then  Av,D  is  the  develop- 
ment of  the  face  avd,  a'v'd'.  With  centre  D  and  radius  dc  obtain  C  on  the  arc  of  radius  v'c"; 
similarly  for  the  remaining  faces,  completing  the  development  vt- — AD...A1. 

The  shaded  area  v2 — TP...T  is  the  development  of  that  part  of  the  pyramid  above  the  oblique 
plane  s'p',  found  by  laying  off,  on  the  various  edges  as  seen  in  the  development,  the  distances  along 
those  edges  from  the  vertex  to  the  cutting  plane;  thus  v^N=v'n',  the  real  length  of  v'm';  vtP  = 
vlpl;  the  length  of  v' p' ;  v^S=v's',  the  only  elevation  showing  actual  length. 

418.  The  development  of  an  oblique  cone.      The   usual   method   of  solving  this  problem  gives  a  result 
which,    although    not    mathematically    exact,   is    a    sufficiently    close 

approximation  for  all  practical  purposes.  In  it  the  cone  is  treated 
as  if  it  were  a  pyramid  of  many  sides.  The  length  of  any 
element  is  then  found  as  in  the  last  problem.  Thus  in  Fig.  254 
an  element  v  c  is  carried  to  v  c"  about  the  vertical  axis  v  o. 

In   Fig.  255   we  have  v'.ag  for  the   elevation   of  the   cone,  and 

»'  o  —  abc...y   for   the    half 

plan.       Make    o  b "  = 
ob;    then   v' b"  is 
the  real  length 

of   the    element    whose    plan    is    o  b.      Similarly, 
c,  d,  e  and   /  are  carried  by  arcs  to  ag  and 
there  joined   with   v'. 

For  the   development   make    vlA 
equal    and    parallel    to    v' a,   and    at 
any    distance  from   it.      With   vl  as 
a  centre  draw  arcs  with  radii  equal 
to  the  true  lengths  of  the  elements; 
then,    as     in     the     pyramid,     make 
A  £  —  arc    a  b  ;     B  C  =  arc   b  c ,   etc. 
The  greater  the  number  of  divisions  011  the  semi  -  circle  a  b...g  the  more  closely  will  the  develop- 
ment  approximate   to   theoretical   exactness. 

419.  The  five  regular  convex  solids,   with   the   forms    of   their 
developments,     are    illustrated     in     Figs.   256-265.       They     have 
already    been    defined    in    Art.  345,   and    that    five   is   their  limit 
as    to    number    is    thus    shown:      The    faces     are   to    be    equal, 
regular  polygons,   and  the    sum   of   the    plane    angles    forming    a 
solid    angle    must    be   less    than    four    right    angles ;    now   as   the 
angles    of   equilateral   triangles    are    60°    we    may    evidently     have 
groups    of   three,   four    or    five    and    not    exceed  the  limit;    with 
squares    there    can    be    groups    of    three    only,    each    90°;     with 
regular    pentagons,    their    interior    angles     being    108°,    groups    of 
three ;   while  hexagons  are    evidently  impracticable,   since  three   of 
their  interior  angles  would  exactly  equal  four  right  angles,  adapt- 
ing them  perfectly — and  only — to  plane  surfaces.     (See  Fig.  131.) 


f-  2S-7. 


OCTAHEDRON. 


ICOSAHEORON. 


THE    FIVE    REGULAR     CONVEX    SOLIDS. 


155 


The  dihedral  angles   between  the   adjacent   faces   of    regular    solids   are   as   follows:     70°  31'  44"   for 
the   tetrahedron;    90°   for  the    cube;    109°  28'  16"   for  the   octahedron;     116°  35' 


54"   for  the   dodecahedron;     and    138°  11'  23"   for  the   icosahedron. 
A   sphere   can   be   inscribed   in   each   regular  solid   and  can   also 
as   readily   be   circumscribed   about   it. 

The   relation   between   d,   the   diameter   of   a  sphere,   and   e,  the 
edge    of    an    inscribed    regular    solid,    is    illustrated    graphically    by 
Fig.  266,   but   may   be   otherwise   expressed   as   follows : 
d  :  e   : :   v'  3  :  v'  2 ;      for  the  cube  d  :  e  : :   V  3  :  1 

"        "     octahedron        d  :  e   '.'.    ^2  :  1;  "       "     dodecahedron  e  =  the  greater  segment    of   the  edge 

of  an   inscribed  cube   when  the  latter  has   been   medially   divided,  that  is,  in   extreme  and  mean  ratio. 


TETRAHEDRON. 

For  the  tetrahedron 


.  2S3. 


OCTAHEDRON. 


For    the    icosahedron     e  =  the 
chord   of  the   arc   whose   tangent  is 
d;    L  e.,    the    chord   of  63°  26'  6". 
Reference  to  Figs.  256-260  and 
the     use     of    a    set     of     cardboard 
models  which  can  readily  be  made 
\    by    means    of    Figs.    261-265    will 
enable    the    student    to    verify    the 


DODECAHEDRON. 


following    statements    as    to    those    ordinary    views    whose    construction    would    naturally    precede    the 
solution  of  problems  relating  to  these  surfaces. 

•.ass.  In   all  but  the  tetrahedron   each 

face  has  an  equal,  opposite,  parallel 
face,  and  except  in  the  cube  such 
faces  have  their  angular  points  alter- 
nating. (See  Figs.  260,  267,  268.) 
The  tetrahedron  projects  as  in 
Fig.  256,  upon  a  plane  that  is  par- 
allel to  either  face. 

The  cube  projects  in  a  square  upon  a  plane  parallel  to  a  face, 
while  on  a  plane  perpendicular  to  a  body  diagonal  it  projects  as 
a  regular  hexagon,  with  lines  joining  three  alternate  vertices  with  the  centre. 

The  octahedron,   which    is    practically  two    equal  square   pyramids   with   a   common   base,   projects   in 

a    square    and    its    diagonals,    upon    a    plane    perpendicular    to    either    body    diagonal;     in    a    rhombus 

and    shorter    diagonal    when    the    plane    is    parallel    to    one    body   diagonal  and   at  45°   with  the   other 

iFiir-  se7. 


ICOSAHEDRON. 


two;    and   (as  in  Fig.  267)    in    a    regular    hexagon    with    inscribed    triangles    (one    dotted)    when    it  is 
projected  upon  a  plane  parallel  to  a  face. 

The  dodecahedron  projects  as  in  Fig.  268  whenever  the  plane  of  projection  is  parallel  to  a  face. 


156 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


•-  37O- 


Fig.  260  represents  the  icosahedron  projected  on  a  plane  parallel  to  a  face,  and  Fig.  269  when  the 

projection -plane  is   perpendicular  to  an   axis. 

420.     The  Developable  Helicoid.     When  the  word  helicoid    is   used  without   qualification    it    is    under- 
stood to  indicate  one    of  the    warped   helicoids,  such    as    is    met    with,  for    example,  in    screws,  spiral 

staircases  and  screw  propellers.  There  is,  however, 
a  developable  helicoid,  and  to  avoid  confusing  it  with 
the  others  its  characteristic  property  is  always  found 
in  its  name.  As  stated  in  Art.  346,  it  is  generated 
by  moving  a  straight  line  tangentially  on  the  ordi- 
nary helix,  which  curve  (Art.  120)  cuts  all  the 
elements  of  a  right  cylinder  at  the  same  angle. 
Fig.  209  illustrates  the  completed  surface  pictorially; 
Fig.  270  shows  one  orthographic  projection,  and  in 
Fig.  271  it  is  seen  in  process  of  generation  by  the 

hypothenuse  of  a  right-angled   triangle  that  rolls  tangentially   on   a   cylinder. 

The  construction  just  mentioned  is  based  on  the  property  of  non-plane  curves  that  at  any  point 

the  curve  and  its   tangent  make  the   same  angle  with  a  given  plane;    if,  therefore,  the  helix,  beginning 

at    a,   crosses    each    element    of   the    cylinder  at    an 

angle     equal    to    obp    in    the    rolling    triangle,    the 

hypothenuse   of   the    latter  will   evidently   move   not 

only  tangent  to  the   cylinder,  but  also  to  the  helix. 
The    following    important     properties     are     also 

illustrated  by  Fig.  271: 

(a)  The  involute*  of  a    helix   and   of   its    hori- 
zontal   projection  are  identical,   since  the   point   b  is 
the  extremity  of  both  the  rolling  lines,  06  and  pb. 

(b)  The  length   of  any  tangent,  as  mb,  is  that 
of  the  helical   arc  m  a  on   which   it  has   rolled. 

(c)  The    horizontal    projection    b  q    of   any  tan- 
gent 6  m  equals  the  rectification  of  an  arc  a  q  which 

is  the  projection  of  the  helical  arc  from  the  initial  point  a  to  the  point  of  tangency  m. 

The  development  of  one  nappe  of  a   helicoid   is  shown  in  Fig.  273.      It  is  merely  the  area  between 
a    circle    and    its    involute;    but    the    radius    p,   of    the    base    circle,    equals    r    sec2  0,t   in   which  r  is 


-.   S171.. 


*  For  full  treatment  of  the  Involute  of  a  circle  refer  to  Arts.  186  and  187. 

t  This  relation  is  due  to  considerations  of  curvature.  At  any  point  of  any  curve  Its  curvature  is  its  rate  of 
departure  from  its  tangent  at  that  point.  Its  radius  of  curvature  is  that  of  the  osculatory  circle  at  that  point.  (Art. 
380.)  Now  from  the  nature  of  the  two  uniform  motions  imposed  upon  a  point  that  generates  a  helix  (Art.  120) 
the  curvature  of  the  latter  must  be  uniform;  and  if  developed  upon  a  plane  by  means  of  its  curvature  it  must 
become  a  circle  —  the  only  plane  curve  of  uniform  curvature.  The  radius  of  the  developed  helix  will,  obviously,  be 
the  radius  of  curvature  of  the  space  helix.  Following  Warren's  method  of  proof  in  establishing  its  value  let 
o,  6  and  c  (Fig.  272)  be  three  equi-distant  points  on  a  helix,  with  6  on  the  foremost  element;  then  a' c'  is  the 
elevation  of  the  circle  containing  these  points.  One  diameter  of  the  circle  a'b'c'  is  projected  at  6'.  It  is  the 
hypothenuse  of  a  right-angled  triangle  having  the  chord  be,  b'c',  for  its  base.  Let  If  be  the  diameter  of  the 
circle  a'b'c';  2r  =  i>d,  that  of  the  cylinder.  Using  capitals  for  points  in  space  we  have  JTC*  =2pX*e;  also  6~c*  — 
2rXbe;  whence,  dividing  like  members  and  substituting  trigonometric  functions  (see  note  p.  31),  we  have 
P  =  rsec2p,  in  which  0  is  the  angle  between  the  line  BC  and  its  projection. 

Let  6  be  the  inclination  of  the  tangent  to  the  helix  at  b'.  If,  now,  both  A  and  C  approach  B,  the  angle 
3  will  approach  9  as  its  limit;  and  when  A,  B  and  C  become  consecutive  points  we  will  have  p  =  rsec2fl  =  the 
radius  of  the  osculatory  circle  =  the  radius  of  curvature. 

For  another  proof,  involving  the  radius  of  curvature  of  an  ellipse,  see  Olivier,  Cours  de  Otomttrie  Descriptive, 
Third  Ed.,  p.  197. 


THE    INTERSECTION    OF    SURFACES. 


157 


the    radius  of  the    cylinder    on    which    the    helix    originally    lay,  and  6    is    the    angle    at    which    the 

helix  crosses  the  elements.  To  de- 
termine p  draw  on  o.n  elevation  of 
the  cylinder,  as  in  Fig.  274,  a  line 
ab,  tangent  to  the  helix  at  its  fore- 
most point,  as  in  that  position  its 
inclination  6  is  seen 
in  actual  size;  then 
from  o,  where  a  6 
crosses  the  extreme 
element,  draw  an  in- 
definite line,  os,  par- 
allel to  c  d,  and  cut  it  at  m  by  a 
line  a  in  that  is  perpendicular  to  a b 
at  its  intersection  with  the  front 
element  ef  of  the  cylinder;  then 
o  m  =  p  =  ?•  sec2  6.  For  we  have 
oa  =  on  sec  0  =  r  sec  0 ;  and  on  (=  r)  :  oa  ::  o  a  :  om;  whence  om  =  r  sec2  6  =  p. 

The  circumference  of  circle  p  equals  ITT  r  sec  6,  the  actual  length  of  the  helix,  as  may  be  seen 
by  developing  the  cylinder  on  which  the  latter  lies.  The  elements  which  were  tangent  to  the  helix 
maintain  the  same  relation  to  the  developed  helix,  and  appear  in  their  true  length  on  the  development. 
The  student  can  make  a  model  of  one  nappe  of  this  surface  by  wrapping  a  sheet  of  Bristol 
board,  shaped  like  Fig.  273,  upon  a  cylinder  of  radius  r  in  the  equation  r  sec2  6  =  p;  or  a  two- 
napped  helicoid  by  superposing  two  equal  circular  rings  of  paper,  binding  them  on  their  inner  edges 
with  gummed  paper,  making  one  radial  cut  through  both  rings,  and  then  twisting  the  inner  edge 
into  a  helix. 


DEVELOPED   HELICOID 


THE    INTERSECTION    OF    SURFACES. 

421.  When  plane-sided  surfaces  intersect,  their  outline  of  interpenetration  is  necessarily  composed  of 
straight   lines;    but  these   not  being,   in  general,   in   one    plane,  form   what  is   called   a  tivisted  or  warped 
polygon;    also   called   a  gauche   polygon. 

422.  If    either    of    two    intersecting    surfaces    is    curved  their    common    line    will    also    be    curved, 
except  under  special   conditions. 

423.  When   one   of   the   surfaces   is   of   uniform   cross   section — as    a    cylinder   or  a  prism — its   end 
view   will   show   whether  the   surfaces   intersect  in   a   continuous   line   or  in  two  separate  ones.     In  Cases 
a,  b,  c,  d  and  g  of  Fig.  275,   where  the  end  view  of  one  surface  either  cuts  but  one  limiting  line    of 
the   other  surface   or  is   tangent  to   one   or  both    of   the    outlines,   the    intersection    will  be  a  continuous 
line.      Two  separate  curves    of  intersection  will  occur  in  the  other  possible  cases,  illustrated  by  e  and 
/,   in   which  the   end   view   of    one    surface    either    crosses    both    the    outlines    of   the   other   or  else  lies 
wholly   between  them.  a  b  c  d  e  f  -8- 

A    cylinder    will     intersect    a    cone    or    another 
cylinder  in  a  plane  curve  if  its  end  view  is  tangent 

to    the    outlines   of   the    other  surface,   as   in   d    and  Fig-.  STB. 

g,    Fig.  275.     Two    cones    may    also    intersect  in    a    plane  curve,  but  as   the   conditions    to  be  met  are 
not  as   readily  illustrated   they   will   be  treated   in   a   special   problem.       (See   Art.  439). 


158 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


424.  In    general,  the    line    of   intersection    of  two    surfaces    is  obtained,  as  stated  in  Art.  379,  by 
passing    one    or    more    auxiliary    surfaces,  usually   planes,  in    such    manner   as  to  cut  some  easily  con- 
structed  sections — as  straight  lines   or  circles — from   each   of  the  given   surfaces;    the   meeting  -  points   of 
the  sections  lying  in  any  auxiliary  surface  will  lie  on  the  line  sought. 

The  application  of  the  principle  just  stated  is  much  simplified  whenever  any  face  of  either 
of  the  surfaces  is  so  situated  that  it  is  projected  in  a  line.  This  case  is  amply  illustrated  in  the 
problems  most  immediately  following.  Fig-,  svs. 

The    beginner    will    save    much    time    if  « L_         Reference  tine 

he   will  letter   each   projection   of  a  point  as      ^ 
soon  as  it  is   determined. 

425.  The  intersection  of  a  vertical  triangular 
prism  by    a    horizontal    square  prism;    also    the 
developments. 

The  vertical  prism  to  be  1-J-"  high  and 
to  have  one  face  parallel  to  V;  bases  equi- 
lateral triangles  of  1"  side. 

The  horizontal  prism  to  be  2"  long,  its 
basal  edges  f",  and  its  faces  inclined  45°  to 
H ;  its  rear  edges  to  be  parallel  to  and 
y  from  the  rear  face  of  the  horizontal 
prism. 

The  elevations  of  the  axes  to  bisect 
each  other. 

Draw  ei  horizontal  and  1"  long  for  the 
plan  of  the  rear  face  of  the  vertical  prism. 
Complete  the  equilateral  triangle  egi  and 
project  to  levels  1£"  apart,  obtaining  <•'/' 


g'h', 


i'j' 


on  the   elevation. 


Construct  an  end  view  g"  i"  j" 
t"  j"    to    represent    the    reference 
transferred. 

The    end 


h  ,  using 
line     e  t, 


view   of   the    horizontal    prism 


is  the  square  a"b"c"d",  having  its  diagonal  horizontal  and 
upper  and  lower  bases  of  the  other  prism,  and  with  its  corner 
from  i"j".  The  plan  and  front  elevation  of  the  horizontal 
rived  from  the  end  view  as  in  preceding  constructions. 

Since  the  lines  eg  and  gi  are  the  plans  of  vertical  faces 
their  intersection  by  the  edges  a,  b,  c  and  d  of  the  hori- 
n,  m,  I,  p,  q,  r — and  project  to  the  elevations  of  the  same 
edge  a  a  meets  the  other  prism  at  o  and  k,  which  project 
o'  and  k'.  Similarly  for  the  remaining  points. 

The  development  of  the  vertical  prism  is  shown  in  the  shaded  rectangle  EJ',  of  length  Sgi  and 
altitude  e' f.  (See  Art.  411).  The  openings  olp1q1r1  and  klllm1nl  are  thus  found:  For  plt  which 
represents  p',  make  GP=gp,  the  true  distance  ofp'  from  g'h';  then  Pp1  =  xp'.  Similarly,  OG  = 
og,  and  Oq,  =  yq. 


midway  between  the 
b"  one -eighth  inch 
prism  are  next  de- 

of  one  prism  we  note 
zontal  prism — as  at 
edges.  Thus  the 
to  the  level  of  a"  at 


THE    INTERSECTION    OF    PLANE-SIDED    SURFACES. 


159 


The    right    half    of   the    horizontal    prism,   o'a'c'q',  is   developed   at  r2b2bsr3  after  the   method  of 
Art.  412. 

426.     The  intersection  of  two  prisms,   one  vertical,  the  other  horizontal,  each  having  an  edge  exterior  to  the 
other.      The  condition  made  will,  as  already  stated  (Art.  423),  make  the  result   a   single  warped  polygon. 
-.  s,T7.  Let   abed,   1"  x  %",  be   the   plan   of  the   vertical  prism,  which 
stands   with   its   broader  faces  at  some  convenient  angle   c  w  g  to  V. 


From   it   construct   the   front  and   side   elevations,  taking  a  reference 
plane  through   d  for  the  latter. 

Let  the    horizontal  prism    be    triangular   (isosceles   section)    one 
face  inclined  45°   to   H;    another   30°   to   H;     the  rear  edge  to   be 
from   that   of  the   vertical   prism. 

Begin  by  locating  g"  one -fifth  inch   from  the   right   edge,   draw 
f" g"  at  45°,   making   it  of  sufficient  length  to   have  /"   exterior  to 

the  other  prism;  then  f'e"  at  30°  to  H, 
terminated  at  e"  by  an  arc  of  centre  g" 
and  radius  g"f";  finally  e" g".  The  edges  e', 
/'  and  g'  of  the  front  elevation  are  then 
projected  from  e",  f"  and  g". 

The  rear  edge  g  in  the  plan  meets  the 
face  « d  at  s,  which  projects  to  s'  on  the 
elevation  of  the  edge  through  g'. 

Moving    forward    from    n,   the   next   edge 
reached,  of  either  solid,  is   «,  of  the  vertical 
prism.      To    ascertain    the    height    at    which 
it    meets    the    other    prism    we    look    to  the 
end   view,   finding    q"    for    the   entrance  and 
t"  for  the   exit.      Being   on   the  way   up   from   g"  to   e"  we   use    q",   reserving    t"   until    we    deal  .with 
the   face   </"/"•      Projecting   q"   over  to   q'   on   edge  a  a'   draw   q' s',   dotted,   since    it   is   on   a  rear    face. 
Eeturning    to    a    and    moving    toward   b   we   next   reach   the   edge   e,   whose  intersection    p  with   a  b 
is  then  projected  to  edge  e'  at  p'  and  joined  with   q'. 

For  the  next  edge,  b,  we 
obtain  o'  from  the  side  eleva- 
tion, projecting  from  the  inter- 
section of  f"  e"  by  the  edge  b". 
Moving  from  b  toward  w, 
projecting  to  the  front  eleva- 
tion from  either  the  plan  or 
the  side  elevation  according  as 
we  are  dealing  with  a  hori- 
zontal edge  or  a  vertical  one, 
we  complete  the  intersection. 

The     development     of    the 
vertical    prism    is    shown    in    Fig.    278.       As    already 

fully   described,   dd,  =  perimeter   abed  in   Fig.   277;    aQ=a'q';    b  0  =  b' o' ;     ax=ax   (of  Fig.  277); 
x  S  =  vertical   distance  of  *'  from   a'c'.   etc. 


160 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


Although    not    required    in    shop    work    the    draughtsman    will   find   it   an   interesting  and   valuable 
sso-  exercise   to   draw  and   shade   either  solid   after  the   removal   of   the   other;     also    to 

draw    the    common    solid.       The    former    is    illustrated    by    Fig.    279 ;     the    latter 
by  Fig.  280. 

427.      The    intersection    of    two     prisms,    one    vertical,    the    other    oblique    but    with 
edges  parallel  to  V. 

Let  abcd....a'r'  (Fig.  281)  be  the  plan  and   elevation   of  the   vertical   prism. 
Let    the    oblique    prism    be    (a)    inclined    30°    to   H;     (b)    have  its   rear  edge 
•fa"  back   of  the  axis   of   the   vertical   prism;     (c)   have  its  faces  inclined   60°   and 
30°   respectively   to   V;  (d)  have  a  rectangular  base   1|"  X  f".  These   conditions   are   fulfilled  as   follows: 
Through    some    point    o'  of   the    edge    e' o'  draw    an   indefinite    line,    o'f,    at    30°   to    H,    for    the 
elevation  of  the   rear   edge,  and  //,   also   indefinite   in   length   at  first  but  •$$"  back   of  s,  for  the  plan. 


ig-.  2S1- 


Nl 

Take  a  reference  plane  MN  through 
s,   and,   as   in   Art.    397    (b),   construct  an 
auxiliary    elevation    on     MN,    transferring    it    so 
that  it  is   seen  as  a  perpendicular  to  o'f,  thus  obtaining 
the  same  view    of    the    prisms    as    would    be    had    if   looking 
in   the   direction   of  the  arrow.     To   construct  this   make   o" f" 
equal    to  TV;    draw    f"i"    at    60°   to    MN,  and    on    it    com- 
plete a  rectangle   of  the    given    dimensions,    after    which    lay 
off  the  points   of  the   pentagonal  prism   at  the   same   distances 
from    MN  in    both    figures.       Project    back,    in    the    direction 
of  the   arrow,   from     /",   g",-  h"  and  i"  to   the  front   elevation, 
and  draw  g' i'  and  the  opposite  base  each  perpendicular  to  o'/'  and  at  equal  distances  each  side  of  o'. 
For  the  intersection  we  get  any   point  n'  on  an   oblique  edge,   as  g',   by   noting  and  projecting  from 


THE    INTERSECTION   OF    PLANE-SIDED    SURFACES. 


161 


n  where  the  plan  gg  meets  the  face  c  d.  For  a  vertical  edge  as  c'  m'  look  to  the  auxiliary 
elevation  of  the  same  edge,  as  c",  getting  I"  and  771"  which  then  project  back  to  I'  and  m'. 

The  development  need  not  again  be  described  in  detail  but  is  left  for  the  student  to  construct! 
with  the  reminder  that  for  the  actual  distance  of  any  corner  of  the  intersection  from  an  edge  of 
either  prism  he  must  look  to  that  projection  which  shows  the  base  of  that  prism  in  its  true  size: 
thus  the  distance  of  I'  from  the  edge  h'  is  h"l". 

428.  The  intersection  of  pyramidal  surfaces  by  lines  and  planes.  The  principle  on  which  the  inter- 
section of  pyramidal  surfaces  by  plane  -sided  or  single  curved  surfaces  would  be  obtained  is  illustrated 
by  Figs.  282  and  283. 

(a)  In   Fig.  282    the    line    a  b,  a'  b',   is   supposed    to   intersect    the  given   pyra- 
mid.      To    ascertain   its    entrance    and    exit    points   we    regard    the    elevation   a'  b' 
as  representing  a  plane  perpendicular  to  V  and  cutting    the   edges  of  the  pyramid. 
Project  m',   where   one   edge    is    cut,   to    771,   on    the    plan    of   the   same   edge.      Ob- 
taining   n    and    o    similarly    we    have    m  n  o    as    the    plan   of    the  section   made  by 
plane  a'b'.     The   plan  ab   meets   mno  at  s  and  t,  the  plans  of  the  points  sought, 
which   then   project   back   to   a'b'  at   s'   and   t'  for  the   elevations. 

As  a  b,  a'b',  might  be  an  edge  of  a  pyramid  or  prism,  or  an  element  of  a 
conical,  cylindrical  or  warped  surface,  the  method  illustrated  is  of  general  appli- 
cability. 

(b)  In   Fig.  283   the   auxiliary   planes   are   taken  vertical,   instead   of  perpendicular  to   V    as    in    the 
last  case. 


The    plane  MN   cuts    a    pyramid.     To   find   where   any   edge  v'  o'  pierces 
the    plane    MN  pass    an    auxiliary    vertical   plane  xz  through  the  edge,  and 
note    x    and   z,  where  it  cuts  the  limits  of  M  N;    project  these  to   x'  and  z' 
on   the   elevations   of   the   same   limits;     draw  x'z',  which    is   the   elevation   of 
the    line    of   intersection    of   the  original    and  auxiliary   planes,  and    note    s', 
where  it  crosses  v'  o'.     Project  s'  back  to  s  on  the  plan  of  v'o'. 
If   a    side   elevation    has   been   drawn,   in  which   the   plane 
in   question   is   seen   as   a  line  M"  N",  the   height  of  the   points 
of  intersection   can  be   obtained  therefrom   directly. 

429.  The  intersection  of  two  quadrangular  pyramids.  In 
Fig.  284  the  pyramid  v.efgh  is  vertical;  altitude  v'  z'  ;  base 
efgh,  having  its  longer  edges  inclined  30°  to  V. 

The  oblique  pyramid.  Let  s'y',  the  axis  of  the  oblique 
pyramid,  be  parallel  to  V  but  inclined  0°  to  H,  and  be  at  some  small  distance  (approximately  v  k) 
in  front  of  the  axis  of  the  vertical  pyramid;  then  sc  will  contain  the  plan  of  the  axis,  and  also  of 
the  diagonally  opposite  edges  sa  and  s  c,  if  we  make—  as  we  may—  the  additional  requirement 
that  a'c',  the  diagonal  of  the  base,  shall  lie  in  the  same  vertical  plane  with  the  axis. 

Instead   of  taking   a   separate   end   view   of   the   oblique   pyramid    we    may    rotate    its    base    on    the 

diagonal  a'c'  so  that  its  foremost  corner  appears  at  b"  and  the  rear  corner  at  d",  whence  b'  and  d'  are 

derived  by   perpendiculars   b"  b'  and   d"  d',   and    then    the   edges  s'  b'  and  s'  d'.      For  the   plans   b  and 

d  use  sc  as  the  trace  of  the  usual  reference  plane,  and  offsets  equal  to  b'  b"  and   d'd",  as   previously. 

The  angle   a'  c'  d',   or   <f>,   is   the   inclination   of  the   shorter   edges   of  the  base  to   V. 

The    intersection.      Without  going    into    a    detailed    construction    for   each    point    of    the    outline    of 

interpenetration    it    may    be    stated    that    each    method    of   the    preceding    article    is    illustrated  in  this 


162 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


problem,  and  that  there  is  no  special  reason  why  either  should  have  a  preference  in  any  case  except 
where  by  properly  choosing  between  them  we  may  avoid  the  intersection  of  two  lines  at  a  very 
acute  angle — a  kind  of  intersection  which  is  always  undesirable. 

In     the    interest     of    clearness 
only  the  visible  lines  of  the  inter- 
section are  indicated  on  the  plan, 
(a)  Auxiliary  plane  perpendicular 
to  V.    To  find  m,  the  intersection 
of   edge   s d    with    the    face    v he, 
take    s'  d'    as    the    trace     of    the 
auxiliary     plane     containing      the 
edge    in    question;     this    cuts    the 
limiting    edges    of  the    face    at  i' 
and    n'  which    then    project    back 
to    the    plans    of   the    edges    at    i 
and  n.      Drawing  ni    we   note  m, 
where    it    crosses    s  d,  and    project 
m  to   m'   on  s' d'.     Had  ni  failed 
to    meet    sd   within    the   limit   of 
the  face  v  h  e   we  would  conclude 
that  our  assumption  that  sd  met 
that  face  was  incorrect,  and  would 
then  proceed  to  test  it  as  to  some 
other  face,  unless  it  was 
evident     on     inspection 
that    the     edge    cleared 
\         the  other  solid  entirely, 
^>,,  as  is  the  case  with  sb, 
'  /      s'b',  in  the  present  in- 
stance.     By    using    s'b' 
as    an    auxiliary     plane 
the  student  will  obtain 
a  graphic   proof  of  fail- 
ure to   intersect. 

(b)  Auxiliary  plane 
vertical.  This  case  is 
illustrated  by  using  vg 

as  the  trace  of  an  auxiliary  vertical  plane  containing  the  edge  vg,v'g'.  Thinking  this  edge  may 
possibly  meet  the  face  s  b  a  we  proceed  to  test  it  on  that  assumption. 

The  plane  vg  crosses  sa  at  I,  and  sb  at  p;  these  project  to  I'  on  s' a'  and  to  p'  on  s'b'; 
then  p'V  meets  v' g'  at  q',  which  is  a  real  instead  of  an  imaginary  intersection  since  it  lies  between 
the  actual  limits  of  the  face  considered.  From  q'  a  vertical  to  vg  gives  q. 

The  order  of  obtaining  and  connecting  the  points.  The  start  may  be  with  any  edge,  but  once 
under  way  the  progress  should  be  uniform,  and  each  point  joined  with  the  preceding  as  soon  as 
obtained.  Two  points  are  connected  only  when  both  lie  on  a  single  face  of  each  pyramid. 


THE    INTERSECTION    OF    SINGLE    CURVED    SURFACES. 


163 


-   S.S5- 


Supposing  that  q'  was  the  point  first  found,  a  look  at  the  plan  would  show  that  the  edge  sa  of 
the  oblique  pyramid  would  be  reached  before  v  h  on  the  other,  and  the  next  auxiliary  plane  would 
therefore  be  passed  through  sa  to  find  u  u' ;  then  would  come  vh  and  s  d.  Running  down  from 
m  on  the  face  s  d  c  we  find  the  positions  such  that  inspection  will  not  avail,  and  the  only  thing  to 
do  is  to  try,  at  random,  either  a  plane  through  v  h  or  one  through  s  c ;  and  so  on  for  the 
remaining  points. 

The  developments.  No  figure  is  furnished  for  these,  as  nearly  all  that  the  student  requires  for 
obtaining  them  has  been  set  forth  in  Art.  396,  Case  6.  The  only  additional  points  to  which  attention 
need  be  called  are  the  cases  where  the  intersection  falls  on  a  face  instead  of  an  edge.  For 
example,  in  developing  the  vertical  pyramid  we  would  find  the  development  of  j'  by  drawing  v'  j', 
prolonging  it  to  o',  and  projecting  the  latter  to  o,  when  fxo  would  be  the  real  distance  to  lay  off 
from  /  on  the  development  of  the  base;  then  laying  off  the  real  length  of  v' j'  on  v' o'  as  seen  in 
the  development  we  would  have  the  point  sought.  Similarly,  for  tt',  draw  vx;  make  v,zx1^=vx, 
and  v 2 v ,  =  altitude  v' e' ;  then  vlx1  is  the  true  length  of  vx  (in  space);  also,  making  v.ft.f=^vt  and 
drawing  t^tt,  we  find  vltl  to  lay  off  in  its  proper  place  on  the  development  of  the  same  face  vfg. 
430.  An  elbow  or  T-joint,  the  intersection  of  two  equal  cylinders  whose  axes  meet.  Taking  up  curved 
surfaces  the  simplest  case  of  intersection  that  can  occur  is  the  one  under  consideration,  and  which 
is  illustrated  by  Fig.  285. 

The  conditions  are  those  stated  in  Art.  423  for  a  plane  intersection, 
which  is  seen  in  a'  b'  and  is  actually  an  ellipse. 

The  vertical  piece  appears  in  plan  as  the  circle  mq.  To  lay  off  the 
equidistant  elements  on  each  cylinder  it  is  only  necessary  to  divide  the 
half  plan  of  one  into  equal  arcs  and  project  the  points  of  division  to  the 
elevation  in  order  to  get  the  full  elements,  and  where  the  latter  meet  a'  b' 
to  draw  the  dotted  elements  on  the  other. 

The  development  of  the  horizontal  cylinder  is  shown  in  the  line -tinted 
figure.  The  curved  boundary,  which  represents  the  developed  ellipse,  is  in 

reality  a  sinusoid- 
(Refer  to  Art.  171). 
The  relation  of 
the  developed  ele- 
ments to  their 
originals,  fully  de- 
scribed in  Art.  120, 

,  .,fi,e^\  is  so  evident  as  to 

«"     \ 

require   no    further 
remark,    except    to 

call  attention  again  to  the  fact  that  their 
distances  apart,  «',/,,  ftgt,  etc.,  equal  the 
rectification  of  the  small  arcs  of  the  plan. 

431.    To  turn  a  right  angle  loith  a  pipe  by 
a    four-piece     elbow.       This     problem     would 
arise   in   carrying  the   blast   pipe   of  a   furnace  around   a   bend.      Except  as   to  the  number  of  pieces  it 
differs   but  slightly   from   the   last   problem.      Instead   of  one  joint   or  curve   of    intersection  there  would 
be   three,   one   less   than   the   number   of  pieces   in  the  pipe.       (Fig.  286). 


T^ig-.  ESS. 


164 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


Let  o  q  a  show  the  size  of  the  cylinders  employed,  and  be  at  the  same  time  the  plan  of  the 
vertical  piece  o's'n'a'.  Until  we  know  where  a'  n'  will  lie  we  have  to  draw  o'  a'  and  s' n'  until 
they  meet  the  elements  from  S'  and  T',  and  get  the  joint  m  M'  as  for  a  two-piece  elbow.  On 
m  M'  produced  take  some  point  v',  use  it  as  a  centre  for  an  arc  t' x  y  t"  tangent  to  the  extreme 
elements;  divide  this  arc,  between  the  tangent  points,  into  as  many  equal  parts  as  there  are  to  be 
joints  in  the  turn;  then  tangents  at  x  and  y — the  intermediate  points  of  division — will  determine 
the  outer  limits  of  the  joints  at  a',  b'  and  J.  Draw  a'v',  finding  n'  by  its  intersection  with  ss'; 
then  n' I'  parallel  to  a'b',  and  similarly  for  the  next  piece. 

The  developments  of  the  smaller  pieces  would  be  equal,  as  also  of  the  larger.  One  only  is 
shown,  laid  out  on  the  developed  right  section  on  v' x.  The  lettering  makes  the  figure  self  -  interpreting. 

432.  The  intersection  of  two  cylinders,  when  each  is  partially  exterior  to  the  other.  The  given  con- 
dition makes  it  evident,  by  Art.  423,  that  a  continuous  non-  plane  curve  will  result. 

Let    one    cylinder    be    ver- 

.. tical.    2"    in    diameter    and    2" 

n  STE  f     "z n — 

high.  This  is  shown  in  half 
plan  in  h  k  I,  and  in  front  and 
side  elevations  between  hori- 
zontals 2"  apart. 

Let  the  second  cylinder  be 
horizontal;  located  midway  be- 
tween the  upper  and  lower 
levels  of  the  other  cylinder; 
its  diameter  f".  On  the  side 
elevation  draw  a  circle  a" b" 
c"  d"  of  £•"  diameter,  locating 
its  centre  midway  between  k"  I" 
and  &1-/V,  and  in  such  posi- 
tion that  a"  shall  be  exterior 
to  k"kt.  The  elevation  of  the 
horizontal  cylinder  is  then  pro- 
jected from  its  end  view,  and 
is  shown  in  part  without  con- 
struction lines. 
The  curve  of  intersection  is  obtained  by  selecting  particular  elements  of  either  cylinder  and  noting 

where  they  meet  the  other  surface. 

The    foremost    element    of   the    vertical    cylinder    is    k...k'n'm'.      Its    side    elevation,   k"  k1}   meets 

the   circle  at   n"  and  m",   which   give  the  levels    of  n'  and   m'  respectively. 

On    the    horizontal    cylinder    the    highest    and    lowest    elements    are   central   on   the   plan   and   meet 

the   vertical   cylinder  at   e,   which   projects   down  to   the   elements   d'  and   b'. 

The    front    and    rear    elements,   c    and    a,   would    be    central    on    the    elevation.      The    vertical    line 

drawn  from   the   intersection   of   element   c  with   the   arc   hkl    gives    the   right-hand   point   of  the   curve 

of  intersection,  at  the  level  of  a'. 

Any   element  as  gx  may   be  taken  at  random,   and   its   elevation    found  in   either   of  the   following 

ways:     (a)    Transfer    gz,  the    distance    of   the    element    from   MN,  to    s" x    on    the  sid.e  elevation,   and 

draw    xg"    and    g"y',    to     which     last    (prolonged)    project    g    at    g';    or    (b)    prolong    gx   to   meet    a 


THE    INTERSECTION   OF    SINGLE    CURVED    SURFACES. 


165 


-.  see. 


semi-circle  on  a  c  at  g'" ;  make  a'y'  =  xg'"  and  draw  y'  g'.  The  same  ordinate  £</'",  if  laid  off 
below  a,  would  obviously  give  the  other  element  which  has  the  same  plan  gx,  and  to  which  g 
projects  to  give  another  point  of  the  desired  curve. 

433.  The  intersection  of  a  vertical  cone  and  horizontal  cylinder.  Let  the  cone  have  an  altitude,  w  w', 
of  4";  diameter  of  base,  3".  (As  the  cylinder  is  entirely  in  front  of  the  axis  of  the  cone,  only  one- 
half  of  the  latter  is  represented.) 

For  the  cylinder  take  a  diameter  of 
£";  length  3^";  axis  parallel  to  V,  |" 
above  the  base  of  the  cone,  and  \"  from 
the  foremost  element.  Draw  ns  parallel  to 
p' r'  and  •§•"  from  it;  also  g'm'  horizontal 
and  f"  from  the  base;  their  intersection 
s  is  the  centre  of  the  circle  a"d'c"m',  of 
$•"  diameter,  which  bears  to  the  element 
p' r'  the  relation  assigned  for  the  cylinder 
to  the  foremost  element;  said  circle  and 
p'ww'  are  thus,  practically,  a  side  elevation 
of  cylinder  and  cone,  superposed  upon  the 
ordinary  view. 

The  dimensions  chosen  were  purposely 
such  as  to  make  one  element  of  the  cone 
tangent  to  the  cylinder,  that  the  curve 
of  intersection  might  cross  itself  and  give 
a  mathematical  "  dbuble  point." 

The  width  d  b,  of  the  plan  of  the 
cylinder,  equals  m'<7'.  The  plan  of  the 
axis  (as  also  of  the  highest  and  lowest 
elements,  «'  and  <•')  will  be  at  a  distance 
«(/'  from  iv.  Any  element  as  x' y' h'  is  shown  in  plan  parallel  to  pq,  and  at  a  distance  from  it  equal 
either  to  h' y'  if  on  the  rear  or  to  h' x'  if  on  the  front. 

The  element  through  v,  on  which  /'  falls,  is  not  drawn  separately  from  bf  in  plan,  since  vf 
and  m' g'  are  so  nearly  equal  to  each  other;  but  /  must  not  be  considered  as  on  the  foremost 
element  of  the  cylinder,  although  it  is  apparently  so  in  the  plan. 

For  the  intersection  pass  auxiliary  horizontal  planes  through  both  surfaces ;  each  will  cut  from 
the  cone  a  circle,  whose  intersection  with  cylinder-elements  in  the  same  plane  will  give  points  sought. 

A  horizontal  plane  through  the  element  a'  would  be  represented  by  a' <>',  and  would  cut  a  circle 
of  radius  o'z'  from  the  cone.  In  plan  such  circle  would  cut  the  element  a  at  point  1,  and  also  at 
a  point  (not  numbered)  symmetrical  to  it  with  respect  to  w  Q.  Similarly,  the  horizontal  plane  through 
the  element  x' h'  cuts  a  circle  of  radius  I' h'  from  the  cone;  in  plan  such  circle  would  meet  the 
elements  x  and  y  in  two  more  points  (5  and  3)  of  the  curve. 

As  the  curve  is  symmetrical  with  respect  to  wQw',  the  construction  lines  are  given  for  one -half 
only,  leaving  the  other  to  illustrate  shaded  effects.  The  small  shaded  portion  of  the  elevation  of  the 
cylinder  is  not  limited  by  the  curve  along  which  it  would  meet  the  cone,  but  by  a  random  curve 
which  just  clears  it  of  the  right-hand  element  of  the  cone. 

434.     To  find  the  diameter  and  inclination  of  a  cylindrical  pipe  that  will  make  an  elbow  with  a  conical 


166 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


pipe  on  a  given  plane  section  of  the  latter.  Let  v  a  b  be  a  vertical  cone,  and  c  d  the  elliptical  plane 
section  on  which  the  cylindrical  piece  is  to  fit.  The  diameter  of  the  desired  cylinder  will  equal 
the  shorter  diameter  of  the  ellipse  c  d.  To  find  this  bisect  cd  at  e;  draw  fh  horizontally 

through  e,  and  on  it  as  a  diameter  draw  the  semi- 
circumference  fg  h;  the  ordinate  eg  is  the  half  width  of 
the  cone,  measured  on  a  perpendicular  to  the  paper  at  e, 
and  is  therefore  the  radius  of  the  desired  cylinder. 


-.  2ss. 


In  Fig.  290,  the  base  NG  equals  twice 
g  e  of  Fig.  289.  At  first  indefinite  perpen- 
diculars are  erected  at  N  and  G,  on  one  of 
which  a  point  C  is  taken  as  a  centre  for 
an  arc  of  radius  equal  to  c  d  in  Fig  289. 
The  angle  <f>  being  thus  determined  is  next 
laid  off  in  Fig.  289  at  c,  and  c  d  N"  G" 
made  the  exact  duplicate  of  CDNG,  com- 
pleting the  solution. 

The  developments  are  obtained  as  in 
Arts.  120  and  191. 

435.     To  determine  the  conical  piece  which 


-.  2QO. 


(a) 


(b) 


will   properly    connect,    two    unequal    cylinders   of  -  circular  section,  whose  axes  are  parallel,  meeting   then^  either 
(a)   in  circles  or   (b)   in  ellipses;     the   planes   of  the  joints  being   parallel.  ^~~ 

(a)  When  the  joints  are  circles.      To  determine   the  conical  frustum   b  e  h  c  prolong   the   elements  e  b 
and   he  to   v ;     develop  the   cone  v . .  .eh  as  in   Art.  418,  and   on   each  element  as   seen  in  the   develop- 
ment lay   off   the   real   distance  from   v   to   the   upper   base    b  c.      Thus   the   element  whose   plan  is  vtk 
is   of  actual  length  vkt  and  cuts    the   upper  base 

at  a  distance  v  n  from  the  vertex,  which  distance 
is  therefore  laid  on  vkt  wherever  the  latter  ap- 
pears on  the  development. 

(b)  When    the    joints    are    ellipses.       Let    the 
elliptical   joints    no  and   qr    be  the  bases    of   the 
conical    piece     q  n  o  r.       To    get   .the    development 
complete    the    cone    by    prolonging   qn  and   or  to 
w ;     prolong    qr    and    drop    a    perpendicular  to  it 
from    w;     find    the  minor  axis    of    the  ellipse   qr 
as   in  the  first  part  of  Art.  434   and   having  con- 
.structed   the   ellipse   proceed   as  in  Art.  418,   since 
in  Fig.   255  the  arc  abc...g  is  merely  a  special 
case  of  an  ellipse. 

436.  The  projections  and  patterns  of  a  bath-tub.  Before  taking  up  more  difficult  problems  in  the 
intersection  of  curved  surfaces  one  of  the  most  ordinary  applications  of  Graphics  is  introduced,  partly 
by  way  of  illustrating  the  fact  that  the  engineer  and  architect  enjoy  no  monopoly  of  practical 
projections. 

In  Fig.  292  the  height  of  the  main  portion  of  the  tub  is  shown  at  a'd'.  Let  it  be  required 
that  the  head  end  of  the  tub  be  a  portion  of  a  vertical  right  cone  whose  base  angle  c'b'a'  equals  the 
flare  of  the  sides,  such  cone  to  terminate  on  a  curve  whose  vertical  projection  is  o'n'z'a'.  Draw 


THE    INTERSECTION    OF    SINGLE    CURVED    SURFACES. 


167 


two  lines,  b' I'  and   c'i',   at   first  indefinite  in  length- and   at  a   distance  a' d'  apart.     Take  a' d'  vertical, 

and  regard   it  not  only  as  the  projection  of  the  elements  of  tangency  of  the  flat  sides  with  the  conical 

end,   but    also    as    the    elevation    of   part    of 

the    axis,     prolonging     it    to     represent    the 

latter.      Use  v,  the   plan   of  the   axis,  as   the 

centre   for  a   semicircle    of    radius   v  c,   whose 

diameter   e  d  is   the   width   of  the   bottom   of 

the  tub.     Project  c  to  c';    make  angle  v' c' d' 

equal     to     the     predetermined     flare     of    the 

sides;     prolong    v' c'    to   b'    and    o' ;     project 

b'    to    b    on    vc    prolonged    and     draw     arc 

a  bin   with   radius    br,   obtaining   am  for  the 

width   of  the   plan   of  the   top. 

The  plan  of  one-half  the  curve  o' n' z' a' 
is  shown  at  onzm  and  is  thus  found: 
Assume  any  element  v'x'y';  prolong  it  to 
z';  obtain  the  plan  vxy  and  project  z'  upon 
it  at  z.  Similarly  for  n  and  as  many  inter- 
mediate points  as  it  might  seem  desirable 
to  obtain. 

Assuming  that  the  foot  of  the  tub  is 
composed  of  an  oblique  cone  whose  section,  his,  with  the  bottom  is  equal  to  ecd,  and  whose  base 
angle  is  h'i'k',  we  project  i  to  i',  draw  i' k'' at  the  given  angle  to  the  base,  project  k'  to  k,  and 
through  the  latter  draw  the  semicircle  rkq  with  radius  br,  obtaining  the"  plan  of  the  upper  base. 

Joining  the  tangent  points   r  and   s,    h  and   q,   we   have   rs  and    hq   as    the    elements    of   tangency 
of  sides   with  end.     Their  elevations  coincide   in   h'l',  which   meets  k' i'  at  v",  whose  plan  is   v,  on  hq. 

-.  2S3. 


The  development.  Fig.  293  is  the  development  of  one-half  of  the  tub.  EM  equals  b'c';  VO 
equals  r'o';  VZ  equals  r'z",  the  true  length  of  r'z',  obtained,  as  in  previous  constructions,  by  car- 
rying z  to  z,,  thence  to  level  of  z'.  Similarly  at  the  other  end.  (Reference  Articles  191,  408,  418.) 

437.     The  intersection   of  a  vertical  cylinder  and  an  oblique  cone,   their  axes  intersecting. 

Let  MBd  and  M'B'P'N'  be  the  projections  of  the  cylinder;  v'.a'b'  and  v.anbm  those  of  the 
cone.  The  axes  meet  o'  at  an  angle  6  which  is  arbitrary. 


168 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


The  ellipse  anbm  is  supposed  to  be  constructed  by  one  of  the  various  methods  employed  when 
the  axes  are  known;  and  in  this  case  we  get  the  length  of  mn  from  a' b'  and  its  position  from  n', 
while  a 6  is  vertically  above  a' b'. 

(a)  Solution  by  auxiliary  vertical  planes.     Any  vertical  plane 
vis    will    cut    elements    from    the    cylinder    at   e    and    /;    also, 
from    the    cone,   elements    which    meet    the    base    at    s    and    t. 
Project  s  and   t  to   «'   and   t',  join    the    latter  with    the    vertex 
v'  and   note   /'  and   e'    (just   below   d')    where    they    cross   the 
vertical    projection    of   the    elements    from   /  and   e;    these  will 
be  points  in  the   desired   curve   of  intersection. 

By  assuming  a  sufficient  number  of  vertical  planes  through 
v  the  entire  curve  can  be  determined. 

(b)  Solution  by  auxiliary  spheres.      If   two    surfaces   of   revo- 
lution   have   a   common   axis   they   will  intersect   each    other    in 
a    circle    whose    plane   is    perpendicular    to    that    axis.*      This 
property   can   be   advantageously   applied   in   problems   of   inter- 
section. 

With  o' — the  intersection  of  the  axes  —  as  a  centre,  we 
may  draw  circles  with  random  radii  o'f,  o'i,  and  let  these 
represent  spheres.  The  sphere  f'g'w  intersects  the  cone  in  the 
circle  /'(/';  the  cylinder  in  the  circle  h'k'.  These  circles  inter- 
sect each  other  at  x  in  a  common  chord  whose  extremities  are 


pq  and    rw,  their   intersection 


points  of  the  curves  sought. 
They  are  both  projected  in  the 
point  x. 

A   second  pair  of  circular 
sections,    lying     on    the    same 
auxiliary    sphere,   are  seen    at 
z    being   another  point  in   the   solution. 


-  see. 


The  point  y  results   from   taking  the   smaller  sphere. 

438.  Intersection  of  a  cylinder  and  cone,  their  axes  not  lying  in 
the  same  plane. 

In  Fig.  295  let  the  cylinder  be  vertical  and  the  cone  oblique, 
the  axis  of  the  latter  being  parallel  to  V  and  inclined  6°  to  H,  and 
also  lying  at  a  distance  x  back  of  the  axis  of  the  cylinder. 

The  auxiliary  surfaces  employed  may  preferably  be  vertical  planes 
through  the  vertex  of  the  cone,  since  each  will  then  cut  elements  from 
both  cylinder  and  cone.  Thus,  vfe  is  the  h.  t,  of  a  vertical  plane 
which  cuts  e  r,  e'v'  from  the  cone,  and  the  vertical  element  through 
/  from  the  cylinder;  these  meet  in  vertical  projection  at  /',  one  point 
of  the  desired  curve.  The  plan  of  the  intersection  obviously  coincides 
with  that  of  the  cylinder. 


*  By  the  definition  of  a  surface  of  revolution  (Art.  340)  any  point  on  it  can  generate  a  circle  about  its  axis.  If,  then,  two 
surfaces  have  the  same  axis,  any  point  common  to  both  surfaces  would  generate  one  and  the  same  circle,  which  must  also  lie 
on  both  surfaces  and  therefore  be  their  line  of  intersection. 


THE    INTERSECTION    OF    SINGLE    CURVED    SURFACES. 


169 


This 


is    one 


ass. 


439.     Conical  elbow;    right  cones  meeting  at  a  given   angle  and  having  an  elliptical  joint. 

of  the  cases   mentioned  in  Art.  423  as   not  admitting   of  illustration 

in   the   same   way  as   when   dealing   with   surfaces   of    uniform   cross 

section,"   but    a    plane     intersection  is   nevertheless  secured    as   with 

cylinders  by  making  the  extreme  elements  of  the  cones  intersect. 
Let  vx  in   Fig.  296  be  the   axis   of  one   of  the  cones.     If  xyz 

is  the  required  angle  between  the  axes    bisect  it  by  the  line    ym, 

and   draw  the   joint   cd  parallel  to  such  bisector.      The  right   cone 

which  is  to  meet  abed    on    cd   must  be  capable    of   being  cut  in 

a  section  equal  to  cd  by  a  plane  making  an  angle  6  with  its  axis, 

and  must  obviously  have  the  same  base  angle  as  the  original  cone; 

since,   however,  the    upper  portion  vdc    of   the    given    cone    fulfills    these  conditions    we    may    employ 

it  instead  of  a  new  cone,  rotating 
it  about  an  axis  pt  which  is  per- 
pendicular to  the  plane  of  the 
ellipse  dc  and  passes  through  its 
centre.  The  point  o,  in  which 
the  axis  vx  meets  the  plane  dc, 
will  then  appear  at  a,  by  making - 
op=ps;  sv',  drawn  parallel  to 
y  z,  will  be  the  new  direction  of 
vo;  and  an  arc  from  centre  d 
with  radius  cv  will  give  v',  which 
is  then  joined  with  d  and  c  to 
complete  the  construction. 

If  the  length  of  the  major 
axis  of  the  elliptical  joint  had 
been  assigned,  as  ef  for  example, 
that  length  would  have  first  been 
laid  off  from  some  point  e  on 
the  extreme  element  and  parallel 
to  ym,  then  from  /  a  parallel  to 
ve,  giving  g  on  vc;  then  gh 
parallel  and  equal  to  ef,  gives 
the  joint  in  its  proper  place. 

440.  Eight  cones  intersecting  in 
a  non- plane  curve;  axes  meeting  at 
an  oblique  angle.  Let  one  cone, 
v'.a'b',  (Fig.  297)  be  vertical;  the 
other,  oblique,  its  axis  meeting 
v' o'  at  an  angle  0. 

The    plane    a' b'    of   the    base 

of  the  vertical  cone  cuts  the  other  cone  in  an  ellipse  whose  longer 
axis  is  e'f.  As  in  Art.  434  determine  g'  h',  the  semi-minor  axis 
of  this  ellipse.  Project  e',  g'  and  /'  up  to  e,  g  and  /;  make 


170 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


ghi  and  gh,  each  equal  to  g' h' ;  then  on  ef  and  hlhj  as  axes  construct  the  ellipse  ehlfli  as 
in  Art.  131.  Tangents  from  v,  to  the  ellipse  complete  the  plan  of  the  oblique  cone. 

(a)  The  curve  of  intersection,  found  by  auxiliary  planes.  In  order  that  each  auxiliary  plane  shall 
contain  an  element  (or  elements)  of  each  cone,  it  must  contain  both  vertices  and  therefore  the  line 
v'r",  which  joins  them;  hence  its  trace  on  the  plane  e' a' b'  must  pass  through  the  trace,  t' t,  of 
such  line  on  that  plane.  Take  tx  as  the  horizontal  trace  of  one  of  these  auxiliary  planes.  It  cuts 
elements  starting  at  i  and  I  on  the  base  of  the  oblique  cone.  One  of  the  elements  cut  from  the  other 
cone  is  v  p,  which  in  vertical  projection  (v'-  p'~)  crosses  the  elevations  of  the  other  elements  at  q'  and 
r',  two  points  of  the  curves  sought.  Since  the  extreme  elements  of  the  cones  are  parallel  to  V  we 
will  have  c'  and  d' — the  intersections  of  their  elevations  —  for  two  more  points  of  the  curve.  Having 

found  other  points  by  repeating  the 
same  process  the  curve  c'q'rd'  is 
drawn  through  them,  and  the  cones 
may  then  be  developed  as  in  Art.  191. 
(b)  Method  by  auxiliary  spheres, 
Since  the  axes  intersect  we  may  use 
auxiliary  spheres  as  in  Case  (b)  of 
Art.  437.  Thus,  with  o' — the  intersec- 
tion of  the  axes  —  as  a  centre,  take  any 
radius  o' k  and  regard  arc  kyz  as  rep- 
resenting a  portion  of  a  sphere  which 
cuts  the  cones  in  k  s  and  y  z.  These 
meet  at  w,  one  point  of  the  curve  of 
intersection  c'  q'  d'. 

441.  Intersecting  cones,  bases  in  the 
same  plane  but  axes  not.  Let  v.kbfg 
and  e.sQhj  be  the  plans  of  the  cones; 
v.'p'd'  and  e.'Q'c'  their  elevations. 

As  argued  in  Case  (a)  of  the  last 
problem,  the  auxiliary  planes  must  con- 
tain the  line  joining  the  vertices ;  their 
H- traces  would  therefore,  in  the  gen- 
eral case,  pass  through  the  trace  of 
that  line  upon  the  plane  of  the  bases; 
but,  in  the  figure,  both  vertices  hav- 
ing been  taken  at  the  same  height 
above  the  bases,  the  line  which  joins 
them  must  be  horizontal,  hence  parallel 
to  the  H- traces  of  the  auxiliaries:  that 
is,  X  Y,  S  T,  Q  R,  etc.,  are  parallel 
to  v  e. 

c-pi  It  happens    that  the   trace   MN  of 

the  foremost  auxiliary  plane  is  tangent  to  both  bases,  hence  contains  but  one  element  of  each  cone 
and  determines  but  one  point  of  the  desired  curve.  These  elements,  a  e  and  6  v,  meet  at  ni  while 
their  elevations  intersect  at  n'. 


THE    INTERSECTION    OF    SINGLE    CURVED    SURFACES.  171 

Each  of  the  other  planes,  except  X  Y,  being  secant  to  both  bases,  will  cut  two  elements  from 
each  cone,  their  mutual  intersections  giving  four  points  of  the  curve  of  interpenetration.  Thus,  in 
plane  0  P,  the  element  0  e  meets  v  k  in  q  and  v  d  in  x,  while  element  h  e  gives  I  and  m  on  the 
same  elements. 

The  plane  X  Y  being  tangent  to  one  base  while  secant  to  the  other  gives  but  two  points  on  the 
curve  sought. 

Order  of  connecting  the  points.  Starting  with  any  plane,  as  MN,  we  may  trace  around  the  bases 
either  to  the  right  or  left.  Choosing  the  former  we  find,  in  the  next  plane,  the  point  h  to  the 
right  of  a  on  one  base,  and  d  similarly  situated  with  respect  to  b  on  the  other ;  therefore  m,  on  he 
and  d  v,  is  the  next  point  to  connect  with  n.  Elements  o  e  and  /  v  give  the  next  point,  then  u  e  and 
g  v  locate  s,  after  which  those  from  j  and  w  give  the  last  before  a  return  movement  on  the  base  of 
the  t'-cone.  As  nothing  new  would  result  from  retracing  the  arc  gfd  we  continue  to  the  left  from 
w,  although  compelled  to  retrace  on  the  other  base,  since  planes  beyond  j  would  not  cut  the  •v-cone. 
The  element  u  e  is  therefore  taken  again,  and  its  intersection  noted  with  an  element  whose  projection 
happens  to  be  so  nearly  coincident  with  v  x  that  the  latter  is  used. 

Continuing  along  arcs  och  and  ikb  we  reach  the  plane  MN  again,  the  curves  ilx  and  qnm 
crossing  each  other  then  at  71 — -the  point  lying  in  that  plane.  Such  point  is  called  a  double  point, 
and  occurs  on  non- plane  curves  of  intersection  at  whatever  point  of  two  intersecting  surfaces  they 
are  found  to  have  a  common  tangent  plane. 

Tracing  to  the  left  from  a  and  to  the  right  from  b  the  elements  0  e  and  d  v  are  reached,  in  the 
plane  OP.  Their  intersection  x  is  joined  with  71  on  one  side  and  with  the  intersection  of  Se  and 
g  v  on  the  other.  Soon  the  tangent  plane  X  Y  is  again  reached  and  a  return  movement  necessitated, 
during  which  the  arc  X  S  Q,  0  a  is  retraced,  while  on  the  other  base  the  counter-clockwise  motion 
is  continued  to  the  initial  point  b,  completing  the  curve. 

Visibility.  The  visible  part  of  the  intersection  in  either  view  must  obviously  be  the  intersection 
of  those  portions  of  the  surfaces  which  would  be  visible  were  they  separate,  but  similarly  situated 
with  respect  to  H  and  V. 

In  plan  the  point  n  lies  on  visible  elements,  and  either  arc  passing  through  it  is  then  visible 
till  it  passes  (becomes  tangent  to,  in  projection)  an  element  of  extreme  contour  as  at  m  or  t,  when 
it  runs  from  the  upper  to  the  under  side  of  the  surface  and  is  concealed  from  view. 

The  point  w  would  be  visible  on  the  0-cone  but  for  the  fact  that  it  is  on  the  under  side  of 
the  e-cone. 

A  similar  method  of  inspection  will  determine  the  visible  portions  of  the  vertical  projection  of 
the  curve,  which  will  not  be  identical  with  those  of  the  plan.  In  fact,  a  curve  wholly  visible  in  one 
view  might  be  entirely  concealed  in  the  other. 

442.  The  intersection  of  a  vertical  cylinder   and  <tn    oblique  cone,   their  axes  in  the  same  plane.      If   in 
Art.  440  the   vertex   v'  were   removed   to   infinity   the    t>-cone    would    become    a    vertical    cylinder;    the 
line  v' v"   would   become  a   vertical   line   through   v" ;     t  would   be   vertically   above   v" ;    but  the  method 
of  solving   would   be   unchanged. 

443.  In    general,   any    method    of   solving    a    problem    relating    to    a    cone    will    apply   with   equal 
facility  to  a  cylinder,  since  one  is  but  a  special   case  of  the   other.      The    line,  so  frequently   used,  that 
passes   through   the   vertex   of  a   cone   in   the   one   problem   is,   in    the    other,   a   parallel    to    the    axis   of 
the  cylinder.     Planes  containing  both  vertices  of  cones  become  planes  parallel  to  both  axes  of  cylinders. 

In  view  of  the  interchangeability  of  these  surfaces  it  is  unnecessary  to  illustrate  by  a  separate 
figure  all  the  possible  variations  of  problems  relating  to  them. 


172 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


444.    Intersection  of  two  cones,   two  pyramids,  or  of  a  cone  and   a   pyramid,  wnen   neither  the  bases  nor 

axes  lie  in  one  plane. 

One  method  of   solving  this  problem    has  been  illustrated  in  Art.  429,  where  the  intersection  was 

found  by  using  auxiliary  planes  that 
were  either  vertical  or  perpendicular  to 
V;  we  may  as  easily,  however,  employ 
the  method  of  the  last  problem,  viz., 
by  taking  auxiliary  planes  so  as  to 
contain  both  vertices.  This  will  be 
illustrated  for  the  problems  announced, 
by  taking  a  cone  and  pyramid ;  and, 
for  convenience,  we  will  locate  the  sur- 
faces so  that  one  of  them  will  be  ver- 
tical, and  the  base  of  the  other  will  be 
perpendicular  to  V,  since  the  problem 
can  always  be  reduced  to  this  form. 

Let  the  cone  v'.a'b',  v.cdB,  (Fig. 
299)  be  vertical,  and  the  pyramid  o'.  r' 
q' p',  o.rqp,  inclined. 

We  will  assume  that  the  projec- 
tions of  the  pyramid  have  been  found 
as  in  preceding  problems,  from  assigned 
data,  using  o  o  2 ,  o '  p ',  (taken  perpen- 
dicular to  the  base  r'  q')  as  the  refer- 
ence line. 

Join  the  vertices  by  the  line  v'  o', 
v  o,  and  prolong  it  to  get  its  traces, 
ss'  and  tt',  upon  the  planes  of  the 
bases.  All  auxiliary  planes  containing 
the  line  vo,  v'  o',  must  intersect  the 
planes  of  the  two  bases  in  lines  pass- 
ing through  such  traces. 

Prolong  r'  q'  to  meet  the  plane 
a'  b'  at  X.  Project  up  from  X,  get- 
ting yz  for  the  plan  of  the  intersection 
of  the  two  bases. 

We  may  assume  any  number  of 
auxiliary  planes,  some  at  random,  but 
others  more  definitely,  as  those  through 
edges  of  the  pyramid  or  tangent  to  the 
cone.  Taking  first  one  through  an 
edge,  as  or,  we  have  trz  for  its  trace 

on    the    pyramid's    base,  then    zs    for    its    trace    on    H.      The    elements  cv    and    dv  which  lie  in  this 

plane    meet    the    edge    o  r    at  e   and   /,    giving    two    points    of   the    curve.      These    project    to    o'  r'   at 

e'  and  /'. 


INTERSECTIONS.— PRACTICAL    ENGINEERING    DESIGNS. 


173 


The  plane  s  y,  tangent  to  the  cone  along  the  element  u  v,  has  the  trace  y  t  on  the  base  of  the 
pyramid,  and  cuts  lines  j  o  and  k  o  from  its  faces.  These  meet  v  u  at  two  more  points  of  the  curve, 
their  elevations  being  found  by  projecting  j  to  j'  and  k  to  k',  drawing  o'j'  and  o'k',  and  noting 
their  intersections  with  v'u'.  To  check  the  accuracy  of  this  construction  for  either  point,  as  I,  draw 
vv1  perpendicular  to  vu  and  equal  to  v'u',  join  t>,  with  w,  and  we  have  in  vv^u  the  rabatment  of 
a  half  section  of  the  cone,  taken  through  the  element  vu  and  the  axis;  then  lllt  parallel  to  vvlf 
will  be  the  height  of  I'  above  the  base  a' b'. 

With  one  exception,  any  auxiliary  plane  between  sy  and  sz  will  give  four  points  of  the  inter- 
section. The  exception  is  the  plane  s  Y,  containing  the  edge  o  q,  and  which,  on  account  of  hap- 
pening to  be  vertical,  requires  the  following  special  construction  if  the  solution  is  made  wholly  on 
the  plan:  Rabat  the  plane  into  H;  the  elements  it  contains  will  then  appear  at  Avt  and  Svs, 
while  the  edge  o  q  will  .be  seen  in  olql  (by  making  ool^o'0,  and  fl^i  =  g'Q);  elements  and  edge 
then  meet  at  J,  and  Nl  which  counter  -  revolve  to  J  and  N.  We  might,  however,  get  elevations 
first,  as  /',  by  the  intersection  of  element  A'v'  with  edge  o'q';  then  /  from  J'. 

In  the  interest  of  clearness  several  lines  are  omitted,  as  of  certain  auxiliary  planes,  hidden  por- 
tions of  the  ellipses,  and  the  curves  in  which  srq  (the  rear  face)  cuts  the  cone.  The  student 
should  supply  these  when  drawing  to  a  larger  scale. 

BRIDGE     POST     CONNECTIONS.  —  GEARING.  —  SPRINGS.  —  BOLTS,     SCREWS     AND     NUTS. 

445.  Detail  of  a  Bridge. —  Upper -Chord  Post  -  Connection. — A  bridge  or  roof  truss  is  an  assemblage 
of  pieces  of  iron  or  wood,  so  connected  that  the  entire  combination  acts  like  a  single  beam.  Figs. 
300  and  301  are  what  are  called  "skeleton  diagrams"  of  bridge  trusses,  each  piece  or  "member"  of 
the  truss  being  represented  by  a  single  line.  A  BCD  and  A'B'C'D'  are  the  trusses  proper,  the 


former  being  for  an  overhead  track  and  the  latter  for  a  roadway  running  through  the  bridge.  In 
each  case  the  upper  part — called  the  upper  chord— (AD,  B'C')  sustains  compression,  and  is  made  of 
"built  beams,"  formed  by  riveting  together  various  plates  and  lengths  of  structural  iron  in  such 
manner  as  to  form  one  practically  continuous  column. 

The   lower  chords   (B  C,  A' D'*)   sustain  tension,   and   are  made   of  bars   of  high   tensile   strength. 

The  members  that  connect  the  chords  are  called  either  ties  or  struts  according  as  the  strain  in 
them  is  tensile  or  compressive.  Collectively  they  form  the  web  of  the  truss. 

In  the  form  of  truss  illustrated  —  which  is  only  one  of  many  which  have  commended  themselves 
to  the  profession  —  the  vertical  pieces  or  "posts,"  Be,  fg,  etc.,  sustain  compression,  and  are  therefore 
"built"  columns.  They  divide  the  trapezoid  into  parts  called  panels,  which  has  given  the  name 
panel  system  to  this  largely  -  employed  arrangement  of  bridge  members. 

All  the  diagonal  members  in  both  figures,  excepting  A' B'  and  C' D',  are  tension  bars  or  rods. 
Bb  and  Cs  are  struts  whose  sole  office  is  to  keep  the  posts  Ab  and  Da  vertical;  said  posts  then 
conveying  to  the  masonry  whatever  weights  are  transmitted  through  the  truss  to  A  and  D  respectively. 


174 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


Fig.  302  is  a  perspective  view  of  the  connection  at  A  between  the  post  A  b,  the  upper  chord 
A  D,  and  the  four  diagonal  bare  that  are  projected  in  A  B.  The  working  drawings  required  for  such 
connection  are  shown  in  the  three  views  on  the  opposite  page.  Three  analogous  views  would  be 
required  for  the  connection  at  the  foot  of  the  same  post. 

When — as   in  the   case   from   which   our   example   is   taken  —  there  are  two  railroad  tracks  overhead, 
ng.  302.  ^6   members   of  the   middle   truss   will   usually  have   different  propor- 

tions from  those  in  the  outside  trusses,  and  a  separate  set  of  three 
views  has  therefore  to  be  made  for  each  of  its  post  connections, 
so  that  the  smallest  number  of  shop  drawings  for  one  such  bridge  — 
after  making  all  allowance  for  the  symmetry  of  the  structure  with 
reference  to  the  central  plane  M  N- — would  consist  of  twenty  such 
groups  of  three  as  are  illustrated  by  Fig.  304. 

The    upper    projections    (Fig.    304)    are    obviously   a   front    and  a 
side    elevation.      The    lower    figure    may  preferably  be    regarded    as   a 
plan   of  the   object   inverted,   since  that  conception   is   somewhat    more 
natural    than    that   of   the    post    in    its    normal    position,    while    the 
draughtsman  lies   on  his   back   and   gazes   up   at  it  from   beneath. 
Fig.   303   shows  the   inverted   plan    on   a   somewhat    smaller    scale,   and,    although    presented  mainly 
to  illustrate  the   contrast  between   views  with   shade  lines   and  without,  contains  one   or  two  serviceable 
dimensions  that  are  omitted   on  the  other  plate. 

446.  General  description.  Referring  to  the  wood  cut  as  well  as  the  orthographic  projections,  we 
find  the  upper  chord  to  be  composed  of  a  long  cover-plate,  18"  x  ^",  riveted  to  the  top  angles  of 
two  vertical  channel  bars  set  back  to  back;  each  channel  being  15"  high  and  weighing  200  pounds 
per  yard.  The  cross  section  of  the  upper  chord  is  shown  in  solid  black,  with  just  enough  space 
intended  between  plate  and  channels  to  show  that  they  are  not  all  in  one  pieqe. 

Perpendicular  to  the  vertical  faces  of  the  channels  and  through  holes  cut  therein  runs  a  cylinder 
called  a  "pin,"  4"  in  diameter  and  21 J" 
"between  shoulders"  (as  marked  on  the 
plan),  that  is,  between  the  planes  where 
the  diameter  is  reduced  and  a  thread 
turned,  on  which  connection  can  be  made 
with  the  corresponding  post  in  the  next 
truss. 

Four  diagonal  bars  are  sustained  by 
the  pin,  the  latter  passing  through  holes, 
called  "eyes,"  in  the  bars.  Two  of  the 
bar  heads  are  between  the  channels. 

Two  plates  are  inserted  between  each 
of  the  outside  bars  and  the  nearest  channel, 
not  only  to  prevent  the  bar  from  touching 
the  angle,  as  at  h,  but  also  to  relieve  the  metal  nearest  the  pin  from  some  of  the  strain. 


The 


longer    plate    mnFE   is    next    the    channel.     The  other,  nmop,  has  a  kind  of  hub  cast  on  it  which 
rounds   up  to  the  bar  head,   as   shown   in   the   side  elevation. 

The   vertical   post  is   made  up   of  an   I-beam   and   two    channels,   as    shown    by  the  black  sections 
on  the  plan. 


UPPER-CHORD    POST-CONNECTION. 


175 


18"x  VIQ'  CC 

)VER   PLATE 

n                      i"  -K" 

[ 

.1 

4"°  PIN.  21?<" 
BETWEEN    SHOULDERS 


-   SO*. 


Railroad  Bridge 
Past    Connection 

Upper   Clinrd, 


When  drawing  the  above  the  student  should  make  horizontal  dividing  lines  in  all  fractions.  A  brush  tint  of  Prussian  blue  for  all  the  metal 
parts  will  enhance  the  appearance  of  the  drawing  very  materially,  but  the  previous  lining  should  be,  obviously,  in  best  waterproof  ink. 
Centre,  dimension,  and  extension  lines  should  be  in  continuous  red  lines,  unless  for  blue  printing,  in  which  case  all  lines  will  ;be  black. 


176 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


Between  the  upper  chord  and  the  top  of  the  post  is  a  three -quarter -inch  plate,  seen  best  on 
the  plans  at  if  It.  It  is  nicked  out  4",  near  the  nuts  K,  so  as  to  clear  the  two  middle  bars  S  which 
conie  between  the  channels. 

A  5 "  X  3 "  angle  -  iron  runs  from  outside  to  outside  of  channels,  and  is  held  by  rivets  and  by 
the  bolts  marked  H.  A  shorter  piece  of  the  same  kind  is  fastened  by  bolts  K  to  the  plate,  and 
by  rivets  to  the  web  of  the  I-beam. 

447.  Hints  as  to  drawing  the  bridge  post  connection.  Draw  the  main  centre  lines  first;  then  the 
plan  and  side  elevations  simultaneously,  as  the  horizontal  centre  line  of  the  plan  represents  the 
same  vertical  reference  -  plane  as  the  vertical  centre  line  of  the  side  elevation,  and  one  spacing  of 
the  dividers  may  be  made  to  do  double  work. 

The  solid  sections  should  be  drawn  first  of  all;  then  the  pins,  bars,  and  cap  plates  of  the  post 
in  the  order  named.  The  parts  already  drawn  should  next  be  represented  on  the  front  elevation  by 


projecting  up  from  the  plan  and  across  from  the  side  view.  The  filler  plates,  mp  and  mF,  with 
their  rivets,  come  next  on  the  front  elevation,  from  which  they  project  to  the  side. 

Next  in  order  draw  the  angle  irons  on  the  front  elevation,  with  their  bolts,  H'  and  K',  and 
project  them  both  across  and  down.  Finally  put  in  all  remaining  rivets,  and  dimension  the  views. 

The  angle  whose  bolts  are  marked  H  terminates  exactly  on  the  edges  of  the  channels,  as  shown 
in  the  wood -cut,  rather  than  as  indicated  in  the  side  elevation. 

448.  Structural  Iron.  In  Figs.  305  and  306  the  forms  of  iron  more  generally  used  in  bridge  and 
house  construction  are  shown  in  cross  section,  and  may  advantageously  be  drawn  on  an  enlarged  scale. 

Treated  as  described  in  Art.   75  they  may  be  worked  up  with  brush   or  pen  like  Fig.   136. 


TOOTHED     GEARING.— HELICAL    SPRINGS. 


177 


--  3O6. 
15'   I-BEAM. 


449.  Toothed    Gearing.      When    two    shafts     are    to    be    rotated    and    a 
constant    velocity    ratio    maintained    between   them,   it   is    customary   to    fix 
upon    them    toothed    wheels     whose    teeth    are    so    proportioned    that    by 
their    sliding    action    upon    each    other    they    produce    the    motion   desired. 
It  is  not  within    the    intended    scope    of   this    work   to  go   at   length    into 
the    theory   of    gearing,   for   which   the   student  is   referred  to   such    special- 
ized treatises   as  those    of   Grant,    Robinson,  MacCord,   Weisbach   and   Willis; 
but   the    draughtsman    will    find    it    to   his    advantage    to    be   familiar  with 
the    following    rapid    method     of    drawing     the    outlines    of    the    teeth     of 
a   spur    wheel,    in   which   a   remarkably    close    approximation    is    made    by 
circular  arcs  to   the   theoretical  involute   outlines   now  so   much   employed. 

450.  CN   (in   Fig.   307)   is   the   radius   of    the    pitch   circle,   that  is,   the 
circle   which   passes   through   the   middle  of  the  working  part   of  the  tooth. 

The  working  outlines  outside  the  pitch  circle  are  called  faces  (fg,  hi), 
while  within  they  are  designated  as  flanks.  The  flanks  are  rounded  off 
into  the  root  circle  by  small  arcs  called  fillets. 

The  limits  of  the  teeth  on  the  addendum  circle,  as  a,  g,  h,  m,  are 
called  their  points. 

On  the  pitch  circle  the  distance  b  i,  between  corresponding  points 
of  consecutive  teeth,  is  called  the  circular  pitch  (usually  denoted  by  P). 

Knowing  the  pitch  and  the  number  (jV)  of  teeth,  the  radius  of  the 
pitch  circle  will  equal  P  x  N  -*-  2  ir. 

As   one  inch  pitch   and  twenty  teeth  are  taken    as  data  for  the  illus- 


tration,  we  have   C'JV=3".18+. 

The  other  proportions  are  also  expressed 
in  terms  of  the  pitch,  a  frequently -used 
system  therefor  being  indicated  on  the 
figure. 

If  i  is  a  'point  through  which  a  tooth 
outline  is  to  pass,  draw  Ci,  and  on  it  as 
a  diameter  describe  the  semi -circumference 
Cs  i.  An  arc  from  centre  «',  with  a  radius 
of  one-fourth  Ci,  will  give  the  centre  s 
of  the  outline  hij. 

Draw  the  "circle  of  centres"  through 
s,  from  centre  C.  Then  with  *  i  in  the 
dividers,  and  from  centre  /  find  q,  which 
use  in  turn  for  arc  gfe,  and  so  continue. 
The  width  of  rim,  vw,  is  often  made, 
by  a  "shop"  rule,  equal  to  three  -  fourths 
the  pitch.  Reuleaux  gives  for  it  the  fol- 
lowing formula:  v  w  =  0.4  P  -f  .12. 

Diametral  pitch  is  very  frequently  used 
instead    of    circular    pitch,    and    is    simply 
451.     Helical  Springs. 


--  3O7. 


Pitch  (P)  =  bi  =  fl 
Depth,  Mv,=.7  P 
Working  Depth,  Mz,= 
Addendum,  M  N,=.3  I? 
Width  of  Tooth,  fi,=£ 
Width  of  Space,  ilf  ft 
Backlashpft 
Clearance,  zv,=  jo 


Jc 

the    number    of   teeth    per    inch    of    pitch -circle    diameter. 
Draw   first    (Fig.   308)  a  central  helix  acfm..T,  as  follows:     Divide  a  a, — 


178 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


which  is  the  pitch,  or  rise  in  one  turn  —  into  any  number  of  equal  parts,  and  the  semi  -circumference 
A  EM  into  half  as  many  equal  divisions;  then  each  point  marked  with,  a  capital  on  the  half  plan 
gives  two  elevations  (denoted  by  the  same  letter  small)  by  a  process  which  is  self-evident. 


Fig.  308. 


.  30S. 


If  the  spring  is  circular  in  cross  -  section  draw 
a  series  of  circles  having  centres  on  the  helix,  and 
whose  diameters  equal  that  of  the  spring;  then 
the  outlines  of  the  spring  will  be  curves  that  are 
tangent  to  the  circles. 

If  the  spring  be- 
small  the  curvature  of 
the  helix  may  be.  ig- 
nored, and  a  series  of 
parallel  straight  lines 
employed  instead,  drawn 
tangent  to  circular  arcs  as  in  Fig.  309. 

The  upper  half  of  the  figure  gives 
the  method  of  construction,  while  the 
lower  shows  the  spring  in  section,  and 
surrounding  a  solid  cylindrical  core. 

452.  Springs  of  rectangular  cross -section. 
Fig.  311  shows  a  spring  of  this  descrip- 
tion, formed  by  moving  the  rectangle  a  b  c  d  helically,  each  point  describ- 
ing a  helix  which  can  be  constructed  as  described  in  the  last  article. 

When  any  considerable  number  of  turns  of  the  same  helix  has  to 
be  drawn  it  will  save  time  if  the  draughtsman  will  shape  a  strip  of 
pear -wood  into  a  templet,  i.e.,  a  piece  whose  outline  conforms  to  a  line 
to  be  drawn  or  an  edge  to  be  cut,  using  it  then  as  a  curved  ruler  to 
guide  his  pen.  This  is  the  preferable  method  for  all  large  work. 

453.  Square -threaded  screws.  If  in- 
stead of  spirally  twisting  a  rectangu- 
lar bar  the  same  kind  of  surface  be 
cut  upon  a  cylinder  of  wood  or  metal, 
we  shall  have  a  square -threaded  screw. 
This  is  illustrated  by  the  upper  part  of 
Fig.  310,  and  its  construction  is  self- 
evident  after  what  has  preceded.  On  a  larger 
ture  of  the  helices  would  have  to  be  indicated. 

The  upper  view  is  an  elevation  of  a  small  double -threaded 
square  screw,  generated  by  winding  two  equal  rectangles  simul- 
taneously around  the  axis. 

The   central   figure   is   an    elevation    of   a    single -threaded    screw, 
figure   is    a    sectional    view    of  the    nut    for    the    single- 


curva- 


The  lower 
threaded 


screw,    and    evidently    presents    a    surface    identical    with 


that  of  the  back   half  'of  the   screw   which   fits   it. 


SCREW-THREADS.— BOLTS. —  NUTS. 


179 


454.     Triangular -threaded  screws. —  United  States  Standard.  — The  proportions  devised  by  Mr.   William 


Other 


Sellers  of  Philadelphia  have  been  so  generally 
adopted  as  to  be  known  as  the  United  States 
Standard.  They  are  given  in  the  table  on  the 
next  page. 

Fig.  312  shows  a  section  of  the  Sellers 
screw.  It  is  blunt  on  the  thread,  and  also  at 
the  root.  The  part  o  p  B  which  is  removed 
.  B  from  the  point  may  be  regarded  as  filled  in 
at  Ns  t.  A  B  being  the  pitch  (P),  the  widths 
op,  $  t,  are  each  one  -  eighth  of  P. 

With  N  equal  to  the  number  of  threads 
per  inch,  and  D  the  outside  diameter  of  the 
screw  or  bolt,  the  value  of  d  —  the  diameter  at 
the  root — may  be  obtained  from  the  formula 
d  =  £>-(!. 299  H-iV). 

proportions  are  as  follows:  The  pitch  is  equal  to 
0.24 A/  ,0+0.625  —  0.175.  The  depth  of  thread  equals  0.65  P.  For  bolts 
and  nuts,  whether  hexagonal  or  square,  the  "width  across  flats,"  or 
shortest  distance  between  parallel  faces,  equals  1.5  D,  plus  one- eighth 
of  an  inch  for  rough  or  unfinished  surfaces,  or  plus  one  -  sixteenth  of 
an  inch  for  "  finished,"  i.  e.,  machined  or  filed  to  smoothness. 

The  depth  of  nut  equals  the  diameter  of  the  bolt,  for  "rough" 
work.  Tables  should  be  consulted  for  the  proportions  of  finished  pieces. 
Fig.  313  is  a  drawing,  to  reduced  scale,  of  a  finished  £"  bolt. 
The  elevations  show  a  bevel  or  chamfer,  such  as  is  usually  given 
to  a  finished  bolt  or  nut.  On  the  plans  this  is  indicated  by  the 
circles  of  diameter  p  q,  the  latter  usually  a  little  more  than  three- 
fourths  of  the  diameter  a  d. 

To    draw    the    lines    resulting    from 
a    view    showing    "width    across    flats," 
chamfer  lines  z  u,  o  i,  at  30° 
to  the  top,  and  cutting  off 
the  desired  amount.     Draw 
circles    on   the    plans,    with 
diameter  equal  to  u  o.     Pro- 
ject p  and   q  to   P    and    Q, 
and    draw   Px  and    Qy  at   30°    to    the 
top.      Make    Nk    on   the    nut    equal    to 
n  y  on  the  head.      On  the  latter  draw  a  parallel    to  P  Q, 
and    as    far    from   it   as   ou  is    from   vi.      The  arcs   limit- 
ing   the    plane    faces  have  their    centres    found  by  "trial 
and  error,"  three  points   of  each  curve  being  known. 

When     drawn     to     a     small     scale     screws    may    be 
represented     by    either     of    the     conventional    methods    illustrated 


chamfering  proceed   thus:    On 

as  that    of  the    nut,  draw  the 

r-  sis. 


HEAD 


BOLT 


NUT 


by    Figs.    314,    315    and    316. 


180 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


DIMENSIONS  OF  BOLTS  AND  NUTS,  UNITED  STATES  STANDARD  (  SELLERS  SYSTEM) 


Proportions  of  Bolt 

Dimensions  of  Nuts 
Rough  and  Finished 

Dimensions  of  Bolt  Heads 
Rough  and  Finished 

Outside 
Diam. 

D 

At  Root  of  Thread 

N= 
Number 
of 
Threads 
per  inch. 

Width  {  f  ) 
of  Flat. 

V  —  V 

T 

Across 

0 

Flats 

Across 

\  w   / 
Corners 

Across 

\  W/ 
Corners 

Depth 

rrrTh- 

of  Nut 

Across 

0 

Flats 

Across 

0 

Corners 

Across 

O 

Corners 

Depth 

fTTTh" 

Diam. 

Area 

of  Head 

R 

F 

R 

R 

R 

F 

R 

F 

R 

R 

R 

F 

1 
4 

.185 

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20 

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ORTHOGRAPHIC    PROJECTION    UPON    A    SINGLE    PLANE. 


241 


CHAPTER    XV. 


AXONOMETRIC     (INCLUDING    ISOMETRIC)     PROJECTION.— ONE -PLANE    DESCRIPTIVE     GEOMETRY. 

621.  When  but  one  plane  of   projection    is    employed    there    are    but    two    applications   of   ortho- 
graphic   projection    having    special    names.      These   are   Axonometric  (known   also   as  Axometric)  Projection, 
and    One- Plane  Descriptive    Geometry   or   Horizontal  Projection. 

AXONOMETRIC     PROJECTION.  —  ISOMETRIC     PROJECTION. 

622.  Axonometric    Projection,    including    its    much  -  employed    special    form    of  Isometric    Projection,  is 
applicable   to   the   representation   of   the    parts   or 

"details"  of  machinery,  bridges  or  other  con- 
structions in  which  the  main  lines  are  in  direc- 
tions that  are  mutually  perpendicular  to  each 
other. 

An  axonometric  drawing  has  a  pictorial 
effect  that  is  obtained  with  much  less  work 
than  is  involved  in  the  construction  of  a  true 
perspective,  yet  which  answers  almost  as  well 
for  the  conveying  of  a  clear  idea  of  what  the 
object  is;  while  it  may  also  be  made  to  serve 
the  additional  purpose  of  a  working  drawing, 
when  occasion  requires. 

623.  Fundamental      Problem. —  To     obtain     the 
orthographic  projection  of  three  mutually  perpendicu- 
lar lines  or  axes,   and   the  scale  of  real  to  projected 
lengths.      Let  a  b,   be  and   b  d  (Fig.   394)    be  the 
projections   of  three    lines   forming    a   solid    right 
angle    at    b.       Let    the    line    a  b    be     inclined    at 
some    given    angle  0  to  the    plane   of   projection. 
Locate   a   vertical   plane    parallel  to  a  b   and   pro- 
ject   the    latter   upon    it    at    a  'b',    at    6°   to   the 
horizontal.      Since    the    plane    of   the    other    two 
axes    is     perpendicular    to     ab,    a'b',    its    traces 
will  be  P'd'R.    (Art.   303). 

In  order  to  find  either  c  or  d  we  need  to 
know  the  inclination  of  the  axis  having  such 
point  for  its  extremity.  Supposing  /3  given 
for  cb,  draw  b' C  at  /J°  to  GL;  project  C  to  c,  and  draw  arc  c^,  centre  b,  obtaining  c. 

Join  a  with  c;    then  a  c  is  the  trace  of   the    plane  of  the    axes  6  a  and  b  c,  and    being    perpen- 
dicular to   the  third  axis   we   may   draw   the  latter  as   the   line   ebd,   making   90°   with   ac. 


242  THEORETICAL    AND    PRACTICAL    GRAPHICS. 

Carry  d  to  dl  about  6;  project  dl  to  D  and  join  the  latter  with  b'.  Then  Db'  is  the  true 
length,  and  b'DL  (or  <£)  the  inclination,  of  the  third  axis,  b  d. 

Lay  off  a'n',  Da'  and  (7£',  each  one  inch.  Their  projected  lengths  on  the  horizontal  are  respec- 
tively a'n,  Ds  and  Ct.  The  latter  are  then  the  lengths,  representative  of  inches,  for  all  lines 
parallel  to  ab,  be  and  b  d  respectively. 

624.  To  make  an  axonometric  projection   of  a  one-inch  cube,   to   the   scale  just   obtained. 
Although   not   absolutely   necessary,  it  is  customary  to   take  one  axis  vertical. 

Taking  the  a  b- axis  vertical,  the  cube  in  Fig.  394  fulfills  the  conditions.  For  BA  equals  a'n; 
B  D"  equals  Ds,  and  EG"  equals  Ct,  while  the  angles  at  B  equal  those  at  6. 

The  light  being  taken  in  the  usual  direction,  i.e.,  parallel  to  the  body  -  diagonal  of  the  cube 
(G" R),  the  shade  lines  indicated  are  those  which  separate  illumined  from  unillumined  surfaces,  and 
are  those  which  could,  therefore,  cast  shadows. 

625.  The  axonometric  projection  of  a  vertical  pyramid,  of  three -fourths -inch  altitude   and  inch -square 
base,  to   the  same  scale  as   the   cube.     The   pyramid   in   Fig.  394   meets  the  requirements,  xwyz  having 
been  made  equal  to   C"BD"X;    while  the   altitude  mM,  rising    from    the    intersection    of   the    diago- 
nals  of  the  base,   equals   three -fourths   a'n,  the   inch  -  representative  for  the   vertical  axis. 

626.  To  draw  curves  in  axonometric  projection,  obtain   first  the  projections   of   their   inscribed   or  cir- 
cumscribed  polygons,   or   of  a   sufficient  number   of    secant  lines;    then    sketch    the    curve    through    the 
points    on    these    new   lines    which    correspond   to   the    points    common   to    the   curves   and  lines   in  the 
original   figure.      This   will   be  illustrated   fully   in   treating  isometric   projection. 

627.  Isometric     Projection. — -Isometric     Drawing.       When     three     mutually      perpendicular    axes    are 
equally  inclined  to  the   plane   of   projection,  they   will    obviously  make    equal    angles    (120°)    with   each 
other   in    projection.      This  relation    led    to   the  name  "isometric,"  implying    equal    measure,  and    also 
obviates  the   necessity   for  making  a  separate   scale   for  each   axis. 

The  advantages  of  this  method  seem  to  have  been  first  brought  out  by  Prof.  Farish  of  England, 
who  presented  a  paper  upon  it  in  1820  before  the  Cambridge  Philosophical  Society. 

628.  In   practice   the  isometric  scale  is  never  used,  but,  as   all   lines   parallel   to  the  axes  are  equally 
foreshortened,  it   is   customary  to  lay  off  their  given  lengths   directly  upon  the  axes   or  their  parallels, 
the  result  showing  relative   position    and    proportion    of   parts    just    as    correctly  as    a    true    projection, 
but    being    then   called    an    isometric  drawing,   to    distinguish    it  from    the    other.      It    would,  obviously, 
be  the  projection  of  a  considerably   larger   object  than  that  from   which   the   dimensions   were  taken. 

Lines   parallel  to  the  axes  are  called  isometric  lines. 

Any  plane  parallel  to,  or  containing  two  isometric  axes,  is  called  an  isometric  plane. 

629.  To    make    an    isometric    drawing   of   a  cube  of  three -quarter -inch  edges. 
Starting  with  the   usual  isometric  centre,    0,   (Fig.   395)   draw    one  axis  vertical, 
and  on  it  lay  off  OA  equal  to  three -fourths   of  an  inch.     00  and  OB  are 
then  drawn  with  the  30° -triangle  as  shown,  made    equal    in    length  to   0  A, 
and   the   figure  completed   by   parallels   to  the  lines   already   drawn. 

One  body -diagonal  of  the  cube  is  perpendicular  to  the  paper  at   0. 

630.  To  draw    circles    and    other    curves  isometrically,   employ  auxiliary   tan- 
gents and  secants,  obtain  their  isometric  representations,  and  sketch  the  curves 
through  the  proper  points. 

In  Fig.  396  we  have  an  isometric  cube,  and  at  MO'P'N  the  square,  which — by  rotation  on  MN 
and  by  an  elongation  of  M P' — becomes  transformed  into  MOPN.  The  circle  of  centre  S'  then 


ISOMETRIC    DRAWING. 


243 


becomes    the    ellipse   of    centre   S,   whose  points   are   obtained  by   means    of   the    four    tangencies   d',  F, 
E  and    G,  and   by  making  gn  equal  to   gn',   hm  equal   to   h'm',   etc. 

631.  The  isometric  circle  may  be  divided  into   parts   corresponding    to   certain    arcs    on    the    original, 
either     (1)    by   drawing    radii   from    <S"  to  MN,  as   those 

through  b',  c',  d',  (which  may  be  equidistant  or  not, 
at  pleasure)  and  getting  their  isometric  representatives, 
which  will  intercept  arcs,  as  b  d',  d'e,  which  are  the 
isometric  views  of  b'd',  d'e';  or  (2)  by  drawing  a 
semicircle  x  i  y  on  the  major  axis  as  a  diameter,  letting 
fall  perpendiculars  to  xy  from  various  points,  and 
noting  the  arcs  as  1-2,  2-3,  that  are  included  between 
them  and  which  correspond  to  the  arcs  ij,  j  k,  origi- 
nally assumed. 

632.  Shade    lines    on    isometric    drawings.      While    not 
universally    adhered    to,    the     conventional     direction    for 
the    rays,    in    isometric    shadow    construction,   is    that    of 
the    body -diagonal    CR    of   the    cube    (Fig.    395).      This 
makes  in   projection   an   angle   of  30°    with    the    horizon- 
tal.     Its   projection   on   an    isometrically  -  horizontal    plane 
— as   that   of   the    top  —  is    a    horizontal   line    CB;    while 
its   projection    C  A,   on   the    isometric    representation    of   a 
vertical  plane,  is   inclined   60  °   to   the   horizontal. 

633.  To  illustrate  the  principles  just  stated   Fig.  397 
is  given,  in   which   all   the  lines   are    isometric,   with   the 

exception   of  Dz  and  its   parallels,   and   ST.      The   drawing    of  non- isometric   lines    will    be   treated    in 

the  next  article,  but  assuming  the 
objects  as  given  whose  shadows  we 
are  about  to  construct,  we  may  start 
with  any  line,  as  Dz. 

The  ray  Dd  is  at  30°.  Its  pro- 
jection dld  is  a  horizontal  through  the 
plan  of  D.  The  ray  and  its  projection 
meet  at  d.  As  the  shadow  begins 
where  the  line  meets  the  plane,  we 
have  z  d  for  the  shadow  of  D  z.  This 
gives  the  direction  for  the  shadow  of 
any  line  parallel  to  Dz,  hence  for  yv, 
which,  however,  soon  runs  into  the 
shadow  of  B  C.  As  b  is  the  intersec- 
tion of  the  ray  Bb  with  its  projection 
&!&,  it  is  the  shadow  of  B,  and  b^b 
that  of  b^B.  Then  bv  is  parallel  to 

B  0,    the    line    casting    the    shadow    being    parallel    to    the    plane    receiving    it. 

In    accordance    with    the    principle    last    stated,  de  is   equal    and    parallel    to   D  E,   and   ef  to   E  F. 

At  /  the  shadow  turns   to  g,  as   the   ray  fF,  run  back,  cuts  M  G  at  /',  and  f'G  casts  the  Jg- shadow. 


244 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


ase. 


Then  gh  equals    G  H ',  and  hhl  is  the   shadow  of  Hh.      The    projection  jm    catches    the    ray  Mm  at 

m.      Then  m/,  equal  to  Mf,  completes  the   construction. 

The  timber,  projecting  from  the  vertical  plane  PQR,   illustrates   the   60° -angle  earlier  mentioned. 

Kk'  being   perpendicular  to  the    vertical    plane,   its   shadow   Kl  is   at   60°   to  the   horizontal,   and   Klk 

is  the  plane  of  rays  containing  said  edge.  Its  horizon- 
tal trace  catches  the  ray  from  k'  at  k.  Then  nk,  the 
shadow  of  n'k',  is  horizontal,  being  the  trace  of  a  ver- 
tical plane  of  rays  on  an  isometrically -horizontal  plane. 
The  construction  of  the  remainder  is  self-evident. 

Letting  S  T  represent  a  small  rod,  oblique  to  isometric 
planes,  assume  any  point  on  it,  as  u;  find  its  plan, 
M,;  take  the  ray  through  u  and  find  its  trace  w.  Then 
Sw  is  the  direction  of  the  shadow  on  the  vertical  plane, 
and  at  r  it  runs  off  the  vertical  and  joins  with  T. 

634.  Timber  framings,  drawn  isometrically,  are  illustrated 
by  Figures  398  and  399.  In  Fig.  398  the  pieces  marked 

A  and  B  show  one  form  of  mortise   and  tenon   joint,  and    are  drawn  with   the  lines  in  the  custom- 
ary   directions    of  isometric    axes. 

The    same    pieces    are   represented 

again   at    C  and   D,   all    the    lines 

having    been    turned    through    an 

angle     of     30°,     so     that     while 

maintaining     the      same     relative 

direction  to   each  other  and  being 

still  truly   isometric,   they   lie   dif- 
ferently in   relation    to    the    edges 

of   the   paper — a   matter    of   little 

importance     when     dealing     with 

comparatively    small     figures,    but 

affecting    the     appearance     of     a 

large   drawing    very  materially. 
635.    Non-isometric  lines. — Angles 

in    isometric   planes.      In    Fig.    399 

a     portion     of    a     cathedral     roof 

truss   is   drawn  isometrically. 

Three    pieces    are    shown    that 

are  not  parallel  to  isometric  lines. 

To    represent    them    correctly    we 

need    to    know     the     real     angles 

made  by  them  with  horizontal   or 

vertical    pieces,  and  use  isometric 

coordinates   or  "offsets"   in   laying 

them   out   on  the   drawing. 

In    the    lower    figure    we    see    at  6   the  actual  angle  of   the  inclined  piece  Mf  to  the  horizontal. 

Offsets,  fl  and  I  C,  to  any  point   0  of   the  inclined  piece,  are  laid  off   in  isometric  directions  at  f'V 


ISOMETRIC    DRAWING.— NON-ISOMETRIC    LINES. 


245 


-iOO. 


and  I'C',  when   C"/T   (or  0')   is  the   isometric   view  of  6.      A    similar    construction,  not    shown,  gave 
the   directions  of  pieces  D  and  D'. 

Much  depends  on  the  choice  of  the  isometric  centre.  Had  N  been  selected  instead  of  B,  the  top 
surfaces  of  the  inclined  pieces  would  have  been  nearly  or  quite  projected  in  straight  lines,  render- 
ing the  drawing  far  less  intelligible. 

The  student  will  notice  that  the  shade  lines  on  Fig.  399  are  located  for  effect,  and  in  violation 
of  the  usual  rule,  it  having  been  found  that  the  best  appearance  results  from  assuming  the  light  in 
such  direction  as  to  make  the  most  shade  lines  fall  centrally  on  the  timbers. 

636.  Non- isometric  lines.  —  Angles  not  in  isometric  planes.  To  draw  lines  not  lying  in  isometric 
planes  requires  the  use  of  three  isometric  offsets.  As  one  of  the  most  frequent  applications  of 
isometric  drawing  is  in  problems  in  stone  cutting,  we  may  take  one  such  to  advantage  in  illustrating 
constructions  of  this  kind. 

Fig.   400  shows   an   arched   passage-way,   in   plan   and   elevation.     The   surface  no,  r'l'n'o'  is   verti- 
tical    as    far    as    n'o',  and    conical    (with    vertex   J,  C")    from 
there    to    n"o".      The    vertical    surface    on    nn    is    tangent    at 
n'  to  the   cylinder  n'f'e'o".     Similarly,   mm  is   vertical   torn', 
and  there   changes   into  the   cylinder  m'g'h'. 

The  radial  bed  b'  g'  is  indicated  on  the  plan  (though  not 
in  full  size)  by  parallel  lines  at  bcfigzb.  The  bed  a'h'  is 
of  the  same  form  as  b'g',  being  symmetrical  with  it. 

In  Fig.  401  we  have  an  enlarged  drawing  of  the  key- 
stone with  the  plan  inverted,  so  that  all  the  faces  of  the 
stone  may  be  correctly  represented  as  seen.  The  isometric 
drawing  is  made  to  correspond,  that  is,  it  represents  the 
stone  after  a  180  °- rotation  about  an  axis  perpendicular  to 
the  paper. 

The     isometric     block     in     which 

fig-  -ioi. 
b' 


the  keystone  can  be 
inscribed  is  shown  in 
dotted  lines,  its  di- 
mensions, derived 
from  the  proj  ections, 

being  length,  AA^aa;  breadth,  AB=a'b';  height,  A0  =  a'p. 
The  top  surface  a'b'  becoming  the  lower  in  the  isometric, 
reverses  the  direction  of  the  lines.  Thus,  a'  is  seen  at  A, 
and  b'  at  B.  To  get  D  make  AU=a'u,  then  UD  —  ud'. 
Make  C  symmetrical  with  D  and  join  with  B,  and  also  D 
with  A.  WQ  equals  w'q',  for  the  ordinate  of  the  middle 
point  of  the  arc. 

DE  is  not  an  isometric  line,  hence  to  reach  E  from  A 
we  make  AT—a't;  Te"--=te',  and  e"E=ay  (the  dis- 
tance of  e  from  the  plane  a  6). 

The  remainder  of  the  construction  is  but  a  duplication  of 
one    or    other   of   the    above    processes. 

The   principle,  that   lines   that   are  parallel   on   the   object   will   also  be  parallel   on  the   drawing,  may 
be   frequently    availed   of  in   the   interest   of  rapid   construction   or   for  a  check  as  to   accuracy. 


246 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


HORIZONTAL     PROJECTION     OR     ONE -PLANE     DESCRIPTIVE     GEOMETRY. 

637.  One -Plane    Descriptive    Geometry    or    Horizontal    Projection    is    a    method    of   using    orthographic 
projections   with   but   one    plane,   the  fundamental   principle  being    that    the    space  -  position    of   a    point 
is   known  if  we   have   its   projection   on   a  plane  and   also  know   its   distance   from   the   plane. 

Thus,  in  Fig.  402,  a  with  the  subscript  7  shows  that  there  is  a  point  A,  vertically  above  a  and 
at  seven  units  distance  from  it.  The  significance  of  63  is  then  evident,  and  to 
show  the  line  in  its  true  length  and  inclination  we  have  merely  to  erect  perpen- 
diculars a  A  and  B  b,  of  seven  and  three  units  respectively,  join  their  extremities, 
and  see  the  line  A  B  in  true  length  and  inclination. 

In  this  system   the   horizontal   plane    alone    is  used;    One -plane    Descriptive    is 
<*J  ~*3    therefore  applied  only  to  constructions  in  which  the    lines    are    mainly  or    entirely 

horizontal,   as  in  the   mapping   of  small  topographical    or   hydrographical    surveys,  in   which   the   curva- 
ture of  the  earth  is  neglected;    also  in    drawing  fortifications,  canals,  etc. 

The  plane  of  projection,  usually  called  the  datum  or  reference  plane,  is  taken,  ordinarily,  below 
all  the  points  that  are  to  be  projected,  although  when  mapping  the  bed  of  a  stream  or  other  body 
of  water  it  is  generally  taken  at  the  water  line,  in  which  case  the  numbers,  called  indices  or  refer- 
ences, show  depths. 

638.  A  horizontal   line    evidently   needs    but    one    index.      This    is    illustrated    in    mapping    contour 
lines,  which  represent  sections  of  the  earth's  surface  by  a  series  of  equidistant  horizontal  planes. 


In  Fig.  403  the  curves  indicate  such  a  series  of  sections  made  by  planes  one  yard,  metre  or 
other  unit  apart,  the  larger  curve  being  assumed  to  lie  in  the  reference  or  datum  plane,  and  there- 
fore having  the  index  zero. 

The   profile  of  a  section   made  by   any   vertical  plane  M  N  would   be   found  by  laying  off — to  any 
assumed  scale    for    vertical    distances — ordinates    from    the    points    where    the    plane    cuts 
the   contours,  giving  each    ordinate    the  same    number   of  units    as   are   in   the   index   of 
the    curve    from    which    it    starts.      Such    a    section    is    shown    in  the  shaded  portion  on 
the  left,  on  a  ground  line  PQ,  which  represents  MN  transferred. 

639.  The  steepness  of  a  plane  or  surface  is  called  its  slope  or  declivity. 
A  line  of  slope  is  the  steepest  that  can  be  drawn  on  the  surface.  A  scale  of 
slope  is  obtained  by  graduating  the  plan  of  a  line  of  slope  so  that  each  unit 
on  the  scale  is  the  projection  of  the  unit's  length  on  the  original  line. 
Thus,  in  Fig.  404,  if  mn  and  o  B  are  horizontal  lines  in  a  plane,  one  hav- 
ing the  index  4  and  the  other  9,  the  point  B  is  evidently  five  units 
above  A,  and  the  five  equal  divisions  between  it  and  A  are  the  projections  of  those  units. 


HORIZONTAL    PROJECTION    OR     ONE-PLANE    DESCRIPTIVE. 


247 


--  -40E. 


The  scale  of  slope  is  often  used  as  a  ground  line  upon  which  to  get  an  edge  view  of  the 
plane.  Thus,  if  B  B'  is  at  90°  to  B  A,  and  its  length  five  units,  then  B' A  is  the  plane,  and  <£ 
is  its  inclination. 

The  scale  of  slope  is  always  made  with  a  double  line,  the  heavier  of  the  two  being  on  the 
left,  ascending  the  plane. 

As  no  exhaustive  treatment  of  this  topic  is  proposed  here,  or,  in  fact,  necessary,  in  view  of 
the  simplicity  of  most  of  the  practical  applications  and  the  self-evident  character  of  the  solutions, 
only  two  or  three  typical  problems  are  presented. 

640.  To  find  the  intersection  of  a  line  and  plane.  Let  a15630  be  the  line,  and  XY  the  plane. 
Draw  horizontal  lines  in  the  plane  at  the  levels  of  the  iiidexed 
points.  These,  through  15  and  30  on  X  F,  meet  horizontal  lines 
through  a  and  6  at  e  and  d;  ed  is  then  the  line  of  intersection 
of  X  Y  and  a  plane  containing  ab;  hence  c  is  the  intersection  of 
the  latter  with  X  Y. 

The   same  point   c  would    have   resulted   if  the    lines    a  e    and   b  d 
had   been   drawn  in  any   other  direction  while   still  remaining  parallel. 

641.     To  obtain    the    line    of  inter- 
section of  two   planes,   draw    two    hori- 
zontals   in    each,   at   the    same    level, 
and  join   their  points  of  intersection. 
In    Fig.    406    we    have    m  n  and 
qn    as    horizontals    at    level    15,    one 
;R       in    each    plane.      Similarly,    xy    and    ys    are    horizontals    at    level    30.      The    planes 

intersect  in  y  n. 

Were  the  scales  of  slope  parallel,  the  planes  would  intersect  in  a  horizontal  line,  one  point  of 
which  could  be  found  by  introducing  a  third  plane,  oblique  to  the  given  planes,  and  getting  its 
intersection  with  each,  then  noting  where  these  lines  of  intersection  met. 

642.  To  find  the  section  of  a  hill  by  a  plane  of  given  slope.  Draw,  as  in  the  problem  of  Art.  640, 
horizontal  lines  in  the  plane,  and  find  their  intersections  with  contours  at  the  same  level.  Thus, 
in  Fig.  403,  the  plane  X  Y  cuts  the  hill  in  the  shaded  section  nearest  it,  whose  outlines  pass 
through  the  points  of  intersection  of  horizontals  10,  20,  30  of  the  plane,  with  the  like  -  numbered 
contour  lines. 


248 


THEORETICAL    AND    PRACTICAL    GRAPHICS. 


CHAPTER    XVI. 


OBLIQUE    OR    CLINOGEAPHIC    PROJECTION.  —  CAVALIEK    PERSPECTIVE.  — CABINET    PROJECTION. 

MILITARY    PERSPECTIVE. 


643.  If   a   figure   be   projected    upon    a   plane   by  a   system   of  parallel   lines   that  are  oblique  to 
the  plane,  the  resulting  figure  is   called  an  oblique  or  clinographic  projection,  the   latter  term   being  more 
frequently   employed    in    the   applications    of   this   method  to   crystallography.      Shadows    of   objects    in 
the  sunlight  are,   practically,   oblique  projections. 

In    Fig.   407,  ABnm  is   a  rectangle  and  mxyn    its   oblique    projection,  the   parallel  projectors  Ax 
and  By  being  inclined  to  the  plane  of  projection. 

644.  When  the  projectors  make  45°  with  the  plane  this  system  is  known  either  as   Cavalier  Perspective, 

Cabinet  Projection  or  Military  Perspective,  the  plane 
of  projection  being  vertical  in  the  case  of  the 
first  two,  and  horizontal  in  the  last. 

645.  Cavalier  Perspective. —  Cabinet  Projection. — 
Military  Perspective.  As  just  stated,  the  projectors 
being  inclined  at  45°  for  the  system  known  by 
the  three  names  above,  we  note  that  in  this 
case  a  line  perpendicular  to  the  plane  of  pro- 
jection, as  A  m  or  B  n  (Fig.  407),  will  have  a 
projection  equal  to  itself.  It  is,  therefore, 
unnecessary  to  draw  the  rays  for  lines  so  situated,  as  the  known  original  lengths  can  be  directly 
laid  out  on  lines  drawn  in  the  assumed  direction  of  projections. 

Let  a  b  c  d .  n  be  a  cube  with  one  face  coinciding  with  the  vertical  plane.  If  the  arrow  m  indi- 
cates one  direction  of  rays  making  45°  with  V,  then  the  ray  hn,  parallel  to  m,  will  give  h  as 
the  projection  of  n,  and  from  what  has  preceded  we  should  have  c  h  equal  to  c  n,  and  analogously 
for  the  remaining  edges,  giving  a  b  c  d .  i  for  the  cavalier  perspective  of  the  cube. 

Similarly,  EKH  is  a  correct  projection  of  the  same  cube  for  another  direction  of  projectors,  and 
we  may  evidently  draw  the  oblique  edges  in  any  other  direction  and  still  have  a  cavalier  perspec- 
tive, by  making  the  projected  line  equal  to  the  original,  when  the  latter  is  perpendicular  to  the 
plane  of  projection. 

646.  Oblique  projection  of  circles.  Were  a  circle  inscribed  in  the  back  face  of  the  cube  D  K  G 
(Fig.  407)  the  projectors  through  the  points  of  the  circle  would  give  an  oblique  cylinder  of  rays, 
whose  intersection  with  the  vertical  plane  DX  would  be  a  circle,  since  parallel  planes  cut  a  cylinder 
in  similar  sections.  We  see,  therefore,  that  the  oblique  projection  of  a  circle  is  itself  circular  when 
the  plane  of  projection  is  parallel  to  that  of  the  circle.  In  any  other  case  the  oblique  projection  of 
a  circle  may  be  found  like  an  isometric  projection  (see  Art.  631),  viz.,  by  drawing  chords  of  the 
circle,  and  tangents,  then  representing  such  auxiliary  lines  in  oblique  view  and  sketching  the  curve 
(now  an  ellipse)  through  the  proper  points.  Fig.  408  illustrates  this  in  full. 


OBLIQUE  PROJECTION.  — CAVALIER  PERSPECTIVE. 


249 


647.  Oblique  projection  is  even  better  adapted  than  isometric  to  the  representation  of  timber  fram- 
ings,  machine   and  bridge  details,   and   other   objects    in    which 

straight  lines — usually  in  mutually  perpendicular  directions — 
predominate,  since  all  angles,  curves,  etc.,  lying  in  planes  par- 
allel to  the  paper,  appear  of  the  same  form  in  projection, 
while  the  relations  of  lines  perpendicular  to  the  paper  are 
preserved  by  a  simple  ratio,  ordinarily  one  of  equality. 

648.  When  the  rays  make  with    the    plane    of   projection 
an    angle    greater    than    45°,    oblique    projections    give    effects 
more    closely   analogous    to    a  true  perspective,   since    the   fore- 
shortening  is    a  closer  approximation   to   that  ordinarily  exist- 
ing  from   a   finite  point   of  view.      This   is   illustrated    by    Fig. 

409,  in  which  an  object  A  B  D  E,  known  to  be  1"  thick,  has  its  depth  represented  as  only  \" 
in  the  second  view,  instead  of  full  size,  as  in  a  cavalier  perspective,  the  front  faces  being  the  same 
size  in  each.  Provided  that  the  scale  of  reduction  were  known,  abcdkf  would  answer  as  well  for 
a  working  drawing  as  a  45° -projection. 

649.  By   way   of  contrast  with   an   isometric   view    the    timber    framing    represented   by   Fig.   398  is 

q'  ic'  b> 


f.  -41O. 


drawn   in   cavalier   perspective   in   Fig.   410.      Reference   may  advantageously   be  made,  at  this   point,   to 
Figs.   44,   45   and   46,   which   are   oblique   views   of  one   form. 

The    keystone    of   the    arch    in    Fig.   400r  whose    isometric    view   is   shown   in   Fig.   401,  appears  in 
oblique   projection   in   Fig.   411;    the   direction   of   lines   not   parallel   to   the    axes    of    the   circumscribing 
prism   being   found   by   "offsets"   that    must    be    taken    in 
axial    directions. 

650.  Shadows,   in  oblique  projection.      As   in   other   pro- 
jections,  the    conventional    direction    for    the   light  is  that  A 
of  the   body -diagonal  of  the   oblique   cube.     The   edges  to 

draw  in  shade  lines  are  obvious  on  inspection.  (Fig.  412). 

651.  An  interesting  application   of  oblique  projection, 
earlier    mentioned,   is    in    the    drawing    of    crystals.      Fig. 

414   illustrates  this,   in  the   representation   of  a   form   common    in    rluorite    and    called    the    tetrahexahe- 
dron,  bounded  by  twenty -four    planes,   each   of  which    fulfills  the  condition  expressed  in  the   formula 


250 


THEORETICAL    AND    PRACTICAL     GRAPHICS. 


co  :  n :  1 ;    that  is,   each   face  is   parallel   to    one    axis,   cuts    another  at   a   unit's   distance,   and  the  third 

at   some   multiple   of  the   unit. 

The    three    axes    in    this    system    are    equal,    and    mutually    per- 
pendicular;   but  their  projected  lengths  are  a  a',  bb',  cc'. 

The  direction  of  projectors  which  was  assumed  to  give  the 
lengths  shown,  was  that  of  EN  in  Fig.  413,  derived  by  turning  the 
perpendicular  CN  through  a  horizontal  angle  CNM  =  18°  26',  and 
then  elevating  it  through  a  vertical  angle  MNS  =  $°  28';  values 
that  are  given  by  Dana  as  well  adapted  to  the  exhibition  of  the 
forms  occurring  in  this  system. 

The    axes    once    established,  if   we    wish    to    construct  on  them  the  form   oo  :  2 : 1,  we  lay  off   on 


each  (extended)  one-half  its  own  (projected)  length;  thus  cc"  and  c'c'"  each  equal  oc'\  bb"  equals 
ob,  etc.  Then  draw  in  light  lines  the  traces  of  the  various  faces  on  the  planes  of  the  axes.  Thus, 
a'b"  and  a"  b  each  represent  the  trace  of  a  plane  cutting  the  c-axis  at  infinity,  and  the  other  axes 
at  either  one  or  two  units  distance;  the  former  intercepting  the  two  units  on  the  b  -axis  and  the  one 
on  the  a -axis,  while  for  a"  b  it  is  exactly  the  reverse.  Through  the  intersection  of  a'b"  and  a"b' 
a  line  is  drawn  parallel  to  the  c-axis,  indefinite  in  length  at  first,  but  determinate  later  by  the 
intersection  with  it  of  other  edges  similarly  found. 

The  student  may  develop  in  the  same  manner  the  forms    oo:3:l;     oo:2:3;     oo:3:4;     oo:4:5. 


APPENDIX 


Table  of  Proportions  of  Standard   Washers Page    II 

Section   of  Standard   Rail,   one   hundred   pounds   to   the   yard       Page  III 

Sectional   View   of  Allen -Richardson   Slide   Valve Page   IV 

Figures    serviceable    for    variations   of   problems   in  Projection,    Sections,    Conversion   of  views 

from   one  system   of   projection    to  another,  Shadows,   Perspective,   etc Page     V 

Note  to   Art.    113,   on  the   Sections   cut   from   the   Annular  Torus   by   a   Bi- tangent   Plane       .  Page    VI 

Note  to   Art.    131,   on   the   Projection   of   a    Circle   in   an    Ellipse Page   VI 

The   Nomenclature   and   Double   Generation   of  Trochoids Pages  VII-XIV 

Alphabets  and   Ornamental   Devices   for   Titles Pages  XV-XXVIII 


Diam. 
of  Bolt. 

Diam.  of 
Washer. 

Thickness 
of  Washer. 

Diam. 
of  Bolt. 

Diam.  of 
Washer. 

Thickness 
of  Washer. 

1 
4 

9. 
16 

3 

32 

li 

0-8- 
Z16 

5_ 
16 

i 

16 

11 
16 

1 
I 

If 

213 
^16 

A 
16 

3 

8 

13 
16 

1 

8 

1} 

4 

5. 
16 

16 

15 
16 

1 
8 

4 

^ 

3. 

8 

1 

4 

J_ 
16 

It 

•ft 

3_ 

8 

9 
16 

£ 

3 

16 

if 

a 

3 

8 

5 

8 

jA 

116 

3 
16 

2 

4 

3 

8 

3 
1 

Is- 
116 

1 
4 

91 
J4 

*ft 

1 
2 

7 
8 

I13 
A16 

1 
4 

•J 

4 

1 
2 

1 

4 

r, 
16 

If 

_9 
516 

1 
2 

1* 

4 

5 
16 

3 

«! 

6io 

1 
2 

TABLE  OF  PROPORTIONS  OF  WASHERS. 


P.    R.    R.   STANDARD   RAIL  SECTION. 

100    LBS.     PER    YARD. 

Draw  the  above  either  full  size  or  enlarged  50%.  In  either  case 
draw  section  lines  in  Prussian  blue,  spacing  not  less  than  one -twentieth 
of  an  inch.  Dimension  lines,  red.  Dimensions  and  arrow  heads,  black. 
Lettering  and  numerals  either  in  Extended  Gothic  or  Reinhardt  Gothic. 


!L  »"  x  i" 


~4^H*-£*i 

^i'H     i 


ALLEN-RICHARDSON   SLIDE   VALVE. 

Draw  either  full  size  or  larger.  Section  lines  in  Prussian  blue,  one -twentieth  of  an  inch  apart. 
Dimension  lines,  red.  Dimensions  and  arrow  heads,  black.  Lettering  and  numerals  either  in 
Extended  Gothic  or  Reinhardt  Gothic. 


PROOF    THAT     A    HI -TANGENT    PLANE    TO    AN    ANNULAR    TORUS   CUTS    IT    IN   TWO    EQUAL   CIRCLES.       (SEE    ART.    113). 

Let  d  m  2  g  and  nlhb  be  the  plans  of 
the  curves  of  section,  d'z'  their  common 
elevation.  The  plane  MN  cuts  the  equa- 
tors of  the  surface  at  the  points  g,  h,  m,  n; 
and  if  the  sections  are  circles  their  diame- 
ters must  obviously  equal  g  m  or  h  n. 

Take  gk  equal  to  one -half  g  m.  Let 
R  denote  the  radius  of  the  generating  circle 
of  the  torus.  Then  gk  =  R  +  oh  =  «'</. 
We  have  also  o  k  =  R. 

Assume  any  horizontal  plane  P  Q,  cut- 
ting the  torus  in  the  circles  e  c  v  and  t  x  y, 
and  the  plane  MN  in  the  line  ev,  e'.  This 
line  gives  e  and  /  on  one  of  the  curves,  y 
and  v  on  the  other.  Their  elevation  is  e'. 

If  e  d  m  is  a  circle  we  must  have  ek  = 
gk  =  a'o'.  Drawee;  make  k  d  parallel  to 
o  a,  and  e  r  perpendicular  to  it.  Then,  as 
the  difference  of  level  of  the  points  k  and  e 
is  seen  at  s'e',  we  will  have  y'Fe'M-  s'e'2 
for  the  true  length  of  the  line  whose  plan 
is  ke;  and  ke'*  +  s'e'*  is  to  equal  gk'1. 

In   the   right  triangle  spk  we   have   k  e'2 


Then     k  e2  +  s'e"'  =  p  k2 


sp2  +  s'e' 


a  o 


The    second    member    becomes 
by    substitution    and    reduction.      For 


pfc*(  =  o'O  employ  [(s'e'2.  a'o"  -  s'e'.2  R'>)  -=-  R2~],  derived  from  triangles  o'n'e'  and  o'a'b';  and  as 
p8  =  rs  —  rp  =  r8  —  R,  we  have  sp'L  =  (r s—  tf)2  =  d/os2  -or''  —  R)''  =  (y'o V2Ta"'"tt/T—  oV2  —  #)2. 
^  and  «V  disappear  by  using  values  derived  from  the  triangle  a'u'c'. 

NOTE    TO    ART.     131,    ON    THE    ELLIPSE    AND    ITS    AUXILIARY    CIRCLES. 

The   relation    of   T  to    t    and    T,    is    thus    shown    analytically:     Eepresenting    lines    by    letters,    let 

0  A(=  OB  =  FC)  =  «;  OC=b;  OF=OF1  =  c,  a  constant  quan- 
tity; OS=x;  ST=y;  STl  =  y';  FT=P;  F[T=?'.  Then  P  +  P' 
—  2  «  =  i/ 1/2  +  (z  +  c)2  -f  I/?/2  -f-  (x  —  c) 2,  which,  after  squaring,  and 
substituting  62  for  «2  —  c\  gives  62x2  +  a2i/2  =  n262,  the  well- 

b'1 
known    equation   to   the    ellipse;    written   also   y*=  —2(a*  —  a;2)  . .  .  (1). 


In    the    circle    AEBK   we    have    OTl  =  r  =  a  = 
';    whence   (y')!  =  a'  —  x2  .  .  .  (2). 


T,2  +  OS* 


=  y  (?y';2  + 

Dividing   (1)   by   (2),   remembering   that   x  is   the   same    for   both 

y  2         b  2 
7",    and    7',   we   have   7-*r\i=;—j'     whence     y:  y'  ::b:  a;     that    is,    the 

\y  )     a 

ordinate   of  the   ellipse   is   to   the    ordinate   of  the   circle   as  the  semi- 
conjugate   axis   is  to    the    semi  -  transverse.      But    in    the    similar    tri- 

angles   rl\SO,    l\Tt,   we   have   ST:   S  T,  :  :  o  t  :  0  T,;    that    is,   y:y'::b:a,   the    relation    just    established 

otherwise   for   a   point   of  an   ellipse. 


THE  NOMENCLATURE  AND  DOUBLE  GENERATION  OF  TROCHOIDS. 


THE   NOMENCLATURE    AND    DOUBLE    GENERATION    OF    TROCHOIDS. 

[The  anomalies  and  inadequateness  of  the  pre-existing  nomenclature  of  trochoidal  curves  led  to  an  attempt  on  the  part 
of  the  writer  to  simplify  the  matter,  and  the  following  paper  is,  in  substance,  that  presented  upon  the  subject  before  the 
American  Association  for  the  Advancement  of  Science,  in  1887.  Two  brief  quotations  from  some  of  the  communications  to 
which  it  led  will  indicate  the  result. 

From  Prof.  Francis  Reuleaux,  Director  of  the  Royal   Polytechnic  Institution,  Berlin: 

"  /  agree  with  pleasure  to  your  discrimination  of  major,  minor  and  medial  hypotrochoids  and  will  in  future  apply  these  novel  designations." 
From  Prof.  Kichard  A.  Proctor,  B.A.,  author  of  Geometry  of  Cycloids,  etc.: 

"  Tour  system  seems  complete  and  satisfactory.  I  was  conscious  that  my  own  suggestions  were  but  partially  corrective  of  the  manifest  anomalies 
in  former  nomenclatures." 

The  final  outcome  of  the  investigation,  as  far  as  technical  terms  are  concerned,  appears  on  page  59,  in  a  tabular  arrange- 
ment suggested  by  that  of  Kennedy,  and  which  is  both  a  modification  and  an  extension  of  his  ingenious  scheme.  The  property 
of  double  generation  of  troehoids,  when  the  tracing-point  is  not  on  the  circumference  of  the  rolling  circle,  is  even  at  present 
writing  not  treated  by  some  authors  of  advanced  text-books  who  nevertheless  emphasize  it  for  the  epi-,  hypo-  and  peri  -  cycloid. 
This  fact,  and  the  importance  of  the  property  both  in  itself  and  as  leading  to  the  solution  of  a  vexed  question,  are  my  main 
reasons  for  introducing  the  paper  here  in  nearly  its  original  length;  although  to  the  student  of  mathematical  tastes  the 
original  demonstration  presented  may  prove  to  be  not  the  least  interesting  feature  of  the  investigation. 

The  demonstrations  alone  might  have  appeared  in  Chapter  V  —  their  rightful  setting  had  this  been  merely  a  treatise  on 
plane  curves,  but  they  would  there  have  unduly  lengthened  an  already  large  division  of  the  work,  while  at  that  point  their 
especial  significance  could  not,  for  the  same  reason,  have  been  sufficiently  shown.] 

That  would  be  an  ideal  nomenclature  in  which,  from  the  etymology  of  the  terms  chosen,  so  clear  an  idea  could  be 
obtained  of  that  which  is  named  as  to  largely  anticipate  definition,  if  not,  indeed,  actually  to  render  it  superfluous.  This 
ideal,  it  need  hardly  be  said,  is  seldom  realized.  As  a  rule  we  meet  with  but  few  self-explanatory  terms,  and  the  greater 
their  lack  of  suggest! veness  the  greater  the  need  of  clear  definition.  Instances  are  not  wanting  of  ill-chosen  terms  and 
even  actual  misnomers  having  become  so  generally  adopted,  in  spite  of  an  occasional  protest,  that  we  can  scarcely  expect 
to  see  them  replaced  by  others  more  appropriate.  Whether  this  be  the  case  or  not,  we  have  a  right  to  expect,  especially 
in  the  exact  sciences,  and  preeminently  in  Mathematics,  such  clearness  and  comprehensiveness  of  definition  as  to  make 
ambiguity  impossible.  But  in  this  we  are  frequently  disappointed,  and  notably  so  in  the  class  of  curves  we  are  to  con- 
sider. 

Toward  the  close  of  the  seventeenth  century  the  mechanician  De  la  Hire  gave  the  name  of  Roulette — or  roll -traced 
curve — to  the  path  of  a  point  in  the  plane  of  a  curve  rolling  upon  any  other  curve  as  a  base.  This  suggestive  term 
has  been  generally  adopted,  and  we  may  expect  its  complementary,  and  equally  self  -  interpreting  term,  Glissette,  to  keep 
it  company  for  all  time. 

By  far  the  most  interesting  and  important  roulettes  are  those  traced  by  points  in  the  plane  of  a  circle  rolling  upon 
another  circle  in  the  same  plane,  such  curves  having  valuable  practical  applications  in  mechanism,  while  their  geometrical 
properties  have  for  centuries  furnished  an  attractive  field  for  investigation  to  mathematicians. 

The  terms  Cycloids  and  Troehoids  have  been  somewhat  indiscriminately  used  as  general  names  for  this  class  of  curves. 
As  far  as  derivation  is  concerned  they  are  equally  appropriate,  the  former  being  from  /ctf/cXos,  circle,  and  eTSos,  form ;  and  the 
latter  from  Tp6%o?,  wheel,  and  efSos.  Preference  has,  however,  been  given  to  the  term  Troehoids  by  several  recent  writers 
on  mathematics  or  mechanism,  among  them  Prof.  E.  H.  Thurston  and  Prof.  De  Volson  "Wood ;  also  Prof.  A.  B.  W. 
Kennedy  of  England,  the  translator  of  Reuleaux'  Theoretische  Kinematitc,  in  which  these  curves  figure  so  largely  as  cen- 
troids.  Adopting  it  for  the  sake  of  aiding  in  establishing  uniformity  in  nomenclature  I  give  the  following  definition : 

If  two  circles  are  tangent,  either  externally  or  internally,  and  while  one  of  them  remains  fixed  the  other  rolls  upon  it 
without  sliding,  the  curve  described  by  any  point  on  a  radius  of  the  rolling  circle,  or  on  a  radius  produced,  will  be  a 
Trochoid. 

Of  these  curves  the  most  interesting,  both  historically  and  for  its  mathematical  properties,  is  the  cycloid,  with  which 
all  are  familiar  as  the  path  of  a  point  on  the  circumference  of  a  circle  which  rolls  upon  a  straight  line,  i.  e.,  the  circle 
of  infinite  radius. 


The  term  "cycloid"  alone,  for  the  locus  described,  is  almost  universally  employed,  although  it  is  occasionally  qualified 
by  the  adjectives  right  or  common. 

Of  almost  equally  general  acceptation,  although  frequently  inappropriate,  are  the  adjectives  curtate  and  prolate,  to 
indicate  trochoidal  curves  traced  by  points  respectively  without  and  within  the  circumference  of  the  rolling  circle  (or 
generator  as  it  will  hereafter  be  termed)  whether  it  roll  upon  a  circle  of  finite  or  infinite  radius. 

As  distinguished  from  curtate  and   prolate   forms   all  the  other  trochoids   are  frequently  called   common. 

Should  the  fixed  circle  (called  either  the  base  or  director)  have  an  infinite  radius,  or,  in  other  words,  be  a  straight 
line,  the  curtate  curve  is  called  by  some  the  curtate  cycloid;  by  others  the  curtate  trochoid;  and  similarly  for  the  prolate 
forms.  Since  uniformity  is  desirable  I  have  adopted  the  terms  which  seem  to  have  in  their  favor  the  greater  number 
of  the  authorities  consulted,  viz.,  curtate  and  prolate  trochoid.  It  should  also  be  further  stated  here,  with  reference  to  this 
word  "trochoid,"  that  it  is  usually  the  termination  of  the  name  of  every  curtate  and  prolate  form  of  trochoidal  curve, 
the  termination  cycloid  indicating  that  the  tracing  point  is  on  the  circumference  of  the  generator. 

With  the  base  a  straight  line  the  curtate  form  consists  of  a  series  of  loops,  while  the  prolate  forms  are  sinuous, 
like  a  wave  line ;  and  the  same  is  frequently  true  when  the  base  is  a  circle  of  finite  radius ;  hence  the  suggestion  of 
Prof.  Clifford  that  the  terms  looped  and  wavy  be  employed  instead  of  curtate  and  prolate.  But  we  shall  see,  as  we 
proceed,  that  they  would  not  be  of  universal  applicability,  and  that,  except  with  a  straight  line  director,  both  curtate 
and  prolate  curves  may  be,  in  form,  looped,  wavy,  or  neither.  And  we  would  all  agree  with  Prof.  Kennedy  that  as 
substitutes  for  these  terms  "Prof.  Cayley's  kru-nodal  and  ac-nodal  hardly  seem  adapted  for  popular  use."  It  is  therefore 
futile  to  attempt  to  secure  a  nomenclature  which  shall,  throughout,  suggest  both  the  form  of  the  locus  and  the  mode 
of  its  construction,  and  we  must  rest  content  if  we  completely  attain  the  latter  desideratum. 

We  have  next  to  consider  the  trochoids  traced  during  the  rolling  of  a  circle  upon  another  circle  of  finite  radius.  At 
this  point  we  find  inadequacy  in  nomenclature,  and  definitions  involving  singular  anomalies.  The  earlier  definitions  have 
been  summarized  as  follows  by  Prof.  K.  A.  Proctor,  in  his  valuable  Geometry  of  Cycloids:  — 

f  epicycloid      ) 

"The     •<  v    is  the  curve  traced   out  by  a  point  in  the  circumference  of  a  circle   which   rolls   without  sliding: 

(  hypocycloid  j 

(  external  ) 

on  a  fixed   circle  in   the   same   plane,   the   two   circles   being  in    4  >    contact." 

(  internal  j 

As  a  specific  example  of  this  class  of  definition  I  quote  the  following  from  a  more  recent  writer:  —  "If  the  gen- 
erating circle  rolls  on  the  circumference  of  a  fixed  circle,  instead  of  on  a  fixed  line,  the  curve  generated  is  called  an 
epicycloid  if  the  rolling  circle  and  the  fixed  circle  are  tangent  externally,  a  hypocycloid  if  they  are  tangent  internally." 
(Byerly,  Differential  Calculus,  1880.) 

In  accordance  with  the  foregoing  definitions  every  epicycloid  is  also  a  hypocycloid,  while  only  some  hypocycloids  are 
epicycloids.  Salmon  (Higher  Plane  Curves,  1879)  makes  the  following  explicit  statement  on  this  point:  —  "The  hypo- 
cycloid,  when  the  radius  of  the  moving  circle  is  greater  than  that  of  the  fixed  circle,  may  also  be  generated  as  an 
epicycloid." 

To  avoid  any   anomaly   Prof.    Proctor   has   presented   the   following   unambiguous   definition :  — 

( epicycloid      ) 

"  An    •!  J.    is   the  curve   traced  out  by   a  point  on  the  circumference  of  a  circle  which  rolls  without  sliding 

(  hypocycloid  j 

(  outside  ") 

on  a  fixed  circle   in   the   same   plane,   the   rolling   circle   touching  the    •<  v    of  the   fixed  circle." 

( inside    j 

This  certainly  does  away  with  all  confusion  between  the  epi-  and  hypo  -curves,  but  we  shall  find  it  inadequate  to 
enable  us,  clearly,  to  make  certain  desirable  distinctions. 

By  some  writers  the  term  external  epicycloid  is  used  when  the  generator  and  director  are  tangent  externally,  and, 
similarly,  internal  epicycloid  when  the  contact  is  internal  and  the  larger  circle  is  rolling.  Instead  of  internal  epicycloid  we 
often  find  external  hypocycloid  used.  It  will  be  sufficient,  with  regard  to  it,  to  quote  the  following  from  Prof.  Proctor :  — 
"  It  has  hitherto  been  usual  to  define  it  (the  hypocycloid)  as  the  curve  obtained  when  either  the  convexity  of  the  rolling 
circle  touches  the  concavity  of  the  fixed  circle,  or  the  concavity  of  the  rolling  circle  touches  the  convexity  of  the  fixed 
circle.  There  is  a  manifest  want  of  symmetry  in  the  resulting  classification,  seeing  that  while  every  epicycloid  is  thus 
regarded  as  an  external  hypocycloid,  no  hypocycloid  can  be  regarded  as  an  internal  epicycloid.  Moreover,  an  external 
hypocycloid  is  in  reality  an  anomaly,  for  the  prefix  'hypo,'  used  in  relation  to  a  closed  figure  like  the  fixed  circle, 
implies  interiorness." 


To  avoid  the  confusion  which  it  is  evident  from  the  foregoing  has  existed,  and  at  the  same  time  to  conform  to  that 
principle  which  is  always  a  safe  one  and  never  more  important  than  in  nomenclature,  viz.,  not  to  use  two  words  where 
one  will  suffice,  I  prefer  reserving  the  term  "  epicycloid "  for  the  case  of  external  tangency,  and  substituting  the  more 
recently  suggested  name  pericycloid  for  both  "internal  epicycloid"  and  "external  hypocycloid. "  The  curtate  and  prolate 
forms  would  then  be  called  peritrochoids.  By  the  use  of  these  names  and  those  to  be  later  presented  we  can  easily 
make  distinctions  which,  without  them,  would  involve  undue  verbiage  in  some  cases,  and,  in  others,  the  use  of  the 
ambiguous  or  inappropriate  terms  to  which  exception  is  taken.  And  the  necessity  for  such  distinctions  frequently  arises, 
especially  in  the  study  of  kinematics  and  machine  design.  Take,  for  example,  problems  like  many  in  the  work  of 
Reuleaux  already  mentioned,  relating  to  the  relative  motion  of  higher  kinematic  pairs  of  elements,  the  centroids  being 
circular  arcs  and  the  point -paths  trochoids.  In  such  cases  we  are  quite  as  much  concerned  with  the  relative  position  of 
the  rolling  and  fixed  circles  as  with  the  form  of  a  point-path.  In  solving  problems  in  gearing  the  same  need  has  been 
felt  of  simple  terms  for  the  trochoidal  profiles  of  the  teeth,  which  should  imply  the  method  of  their  generation. 

Although  they  have  not,  as  yet,  come  into  general  use,  the  names  pericycloid  and  peritrochoid  appear  in  the  more 
recent  editions  of  Weisbach  and  Reuleaux,  and  will  undoubtedly  eventually  meet  with  universal  acceptance. 

Yet  strong  objection  has  been  made  to  the  term  "  pericycloid "  by  no  less  an  authority  than  the  late  eminent 
mathematician,  Prof.  W.  K.  Clifford,  who  nevertheless  adopted  the  "peritrochoid."  I  quote  the  following  from  his  Elements 
of  Dynamic:  —  "Two  circles  may  touch  each  other  so  that  each  is  outside  the  other,  or  so  that  one  includes  the  other. 
In  the  former  case,  if  one  circle  rolls  upon  the  other,  the  curves  traced  are  called  epicycloids  and  epitrochoids.  In  the 
latter  case,  if  the  inner  circle  roll  on  the  outer,  the  curves  are  hypocycloids  and  hypotrochoids,  but  if  the  outer  circle 
roll  on  the  inner,  the  curves  are  epicycloids  and  peritrochoids.  We  do  not  want  the  name  pericycloids,  because,  as  will 
be  seen,  every  pericycloid  is  also  an  epicycloid;  but  there  are  three  distinct  kinds  of  trochoidal  curves."  As  it  will 
later  be  shown  that  every  peri  -  trochoid  can  also  he  generated  as  an  epi -trochoid  we  can  scarcely  escape  the  conclusion 
that  the  name  peritrochoid  would  also  have  been  rejected  by  Prof.  Clifford,  had  he  been  familiar  with  this  property  of 
double  generation  as  belonging  to  the  curtate  and  prolate  forms  as  well.  But  it  is  this  very  property,  possessed  also  by 
the  hypo- trochoids,  which  necessitates  a  more  extended  nomenclature  than  that  heretofore  existing,  and  I  am  not  aware 
that  there  has  been  any  attempt  to  provide  the  nine  terms  essential  to  its  completeness.  These  it  is  my  principal  object 
to  present,  and  that  they  have  not  before  been  suggested  I  attribute  to  the  fact  that  the  double  generation  of  curtate 
and  prolate  trochoidal  curves  does  not  seem  to  have  been  generally  known,  being  entirely  ignored  in  many  treatises 
which  make  quite  prominent  the  fact  that  it  is  a  property  of  the  epi-  and  hypo -cycloids,  while,  as  far  as  I  have  seen, 
the  only  writer  who  mentions  it  proves  it  indirectly,  by  showing  the  identity  of  trochoids  with  epicyclics  and  establishing 
it  for  the  latter. 

As  it  is  upon  this  peculiar  and  interesting  feature  that  the  nomenclature,  as  now  extended,  depends,  the  demonstra- 
tions necessary  to  establish  it  are  next  in  order. 

For  the  epi-  and  hypo -cycloid  probably  the  simplest  method  of  proof  is  that  based  upon  the  instantaneous  centre, 
and  which  we  may  call  a  kinematic,  as  distinguished  from  a  strictly  geometrical,  demonstration.  It  is  as  follows : — 

Let  F  (Figs.  1  and  2)  be  the  centre  of 
the  fixed  circle,  and  r  that  of  a  rolling  circle, 
the  tracing  point,  P,  being  on  the  circum- 
ference of  the  latter.  The  point  of  contact, 
g,  is  —  at  the  moment  that  the  circles  are  in 
the  relative  position  indicated — an  instanta- 
neous centre  of  rotation  for  every  point  in 
the  plane  of  the  rolling  circle;  the  line  Py, 
joining  such  point  of  contact  with  the  tracing 
point,  is  therefore  a  normal  to  the  trochoid 
that  the  point  P  is  tracing.  But  if  the 
normal  Py  be  produced  to  intersect  the  fixed  circle  in  a  second  point,  Q,  it  is  evident  that  the  same  infinitesimal  arc 
of  the  trochoid  would  be  described  with  Q  serving  as  instantaneous  centre  as  when  q  fulfilled  that  office.  The  point 
P  will,  therefore,  evidently  trace  the  same  curve,  whether  it  be  considered  as  in  the  circumference  of  the  circle  r,  or  in 
that  of  a  second  and  larger  circle,  R,  tangent  to  the  fixed  circle  at  Q. 


It  is  worth   while,  in   this  connection,  to  note  what  erroneous  ideas   with   regard  to   these  same  loci  were  held  by  some 
writers  as  late   as  the  middle    of   this    century,  —  ideas   whose  falsity  it  would    seem  as    if   the    most  elementary  geometrical 

construction  would  have  exposed.  Reuleaux  instances  the  following  statement 
made  by  "Weissenborn  in  his  Cyclischen  Kurven  (1856)  :  "  If  the  circle 
described  about  ma  roll  upon  that  described  about  M,  and  if  the  describing 


point,   B0,    describe    the    curve 


as    the    inner  circle   rolls    upon   the 


arc  B0i,  then,  evidently,  if  the  smaller  circle  be  fixed  and  the  larger  one 
rolled  upon  it  in  a  direction  opposite  to  that  of  the  former  rotation,  the 
point  of  the  great  circle  which  at  the  beginning  of  the  operation  coincided 
with  B0  describes  the  same  line  BoPjPj."  The  fallacy  of  this  statement 
is  to  us,  perhaps,  in  the  light  of  what  has  preceded,  a  little  more  evident 
than  "Weissenborn  's  deduction  ;  although,  as  Reuleaux  says,  "his  '  evidently' 
expresses  the  usual  notion,  and  the  one  which  is  suggested  by  a  hasty 
pre-judgment  of  the  case.  In  point  of  fact  B0  describes  the  pericycloid 
BoE'E",  which  certainly  differs  sufficiently  from  the  hypocycloid  B,,?!?.,." 
We  have  next  to  consider  the  curtate  and  prolate  epi-,  hypo-  and 
peri  -  trochoids. 

As  previously  stated,  I  have  seen  no  direct  proof  that  they  also  possess 
the  same  property  of  double  generation,  but  find  that  the  kinematic  method  lends  itself  with  equal  readiness  to  its 
demonstration. 

For  the  hypotrochoids,  let  R,  Fig.  4,  be  the  centre  of  the  first  rolling  circle  or  generator,  F  that  of  the  first  director, 
and  P  the  initial  position  of  the  tracing  point.  The  initial  point  of  tangency  of  generator  and  director  is  m.  Let  the 
generator  roll  over  any  arc  of  the  director,  as  m  Q.  The  centre  R  will  then  be  found  at  E,,  and  the  tracing  point  P 
at  P2.  The  point  of  contact,  Q,  will  then  be  the  instantaneous  centre  of  rotation  for  P2,  and  P2  Q  will,  therefore,  be 
a  normal  to  the  trochoid  for  that  particular  position  of  the  tracing  point. 

The  motion  of  P  is  evidently  circular  about  R,  while  that  of  R  is  in  a  circle  about  F.  The  curve  P  P,  P2  .....  P6 
is  that  portion  of  the  hypotrochoid  which  is  described  while  P  describes  an  arc  of  180°  about  R,  the  latter  meanwhile 
moving  through  an  arc  of  108°  about  F,  the  ratio  of  the  radii  being  3:5. 

Now  while  tracing  the  curve  indicated  the  point  P  can  be  considered  as  rigidly  connected  with  a  second  point,  p, 
about  which  it  also  describes  a  circle,  p  meanwhile  (like  R)  describing  a  circle  about  F.  Such  a  point  may  be  found 
as  follows:  —  Take  any  position  of  P,  as  P2,  and  join  it  with  the  corresponding  position  of  R,  as  R2;  also  join  R2  to 
F.  Let  us  then  suppose  P2R  and  R2F  to  be  adjacent  links  of  a  four-link  mechanism.  Let  the  remaining  links,  Fp2 
and  p2P2,  be  parallel  and  equal  to  P2R2  and  R2F  respectively.  Taking  F  as  the  fixed  point  of  the  mechanism  let  us 
suppose  P2  moved  toward  it  over  the  path  P2  P3  .  .  .  .  P6.  Both  R2  and  p2  will  evidently  describe  circular  arcs  about  F; 
while  the  motion  of  P2  with  respect  to  p2  will  be  in  a  circular  arc  of  radius  p2P2.  We  may,  therefore,  with  equal 
correctness,  consider  p2  as  the  centre  of  a  generator  carrying  the  point  P2,  and  p2F  a  new  line  of  centres,  intersected  by 
the  normal  P2  Q  in  a  second  instantaneous  centre,  g,  'which,  in  strictest  analogy  with  Q,  divides  the  line  of  centres  on 
which  it  lies  into  segments,  p2q  and  F  q,  which  are  the  radii  of  the  second  generator  and  director  respectively;  q  being, 
like  Q,  the  point  of  contact  of  the  rolling  and  fixed  circles  for  the  instant  that  the  tracing  point  is  at  P,  .  The  second 
generator  and  director,  having  p^q  and  qF  respectively  for  their  radii,  are  represented  in  their  initial  positions,  p  being 
the  centre  of  the  former,  and  p.  the  initial  point  of  contact.  The  second  generator  rolls  in  the  opposite  direction  to 
the  first. 

It  is  important  to  notice  that  whereas  the  tracing  point  is  in  the  first  case  within  the  generator  and  therefore  traces 
the  curve  as  a  prolate  hypotrochoid,  it  is  without  the  second  generator  and  describes  the  same  curve  as  a  curtate  hypo- 
trochoid. If  we  now  let  R  and  F  denote  no  longer  the  centres,  but  the  radii,  of  the  rolling  and  fixed  circles,  respec- 
tively, we  have  for  the  first  generator  and  director  2  R  >  F,  and  for  the  second  2  R  <  F. 

It  occurred  to  me  that  a  distinction  could  very  easily  be  made  between  trochoids  generated  under  these  two 
opposite  relations  of  radii,  by  using  the  simple  and  suggestive  term  major  hypotrochoid  when  2  R  is  greater  than  F,  and 
minor  hypotrochoid  when  the  opposite  relation  prevails.  We  would  then  say  that  the  preceding  demonstration  had  estab- 
lished the  identity  of  a  major  prolate  with  a  minor  curtate  hypotrochoid. 

Similarly   the  identity  of  major  curtate  and  minor  prolate  forms  could  be  shown. 


If  the  tracing  point  were  on  the  circumference  of  the  generator  the  trochoids  traced  would  be,  by  the  new  nomen- 
clature, major  and  minor  \\ypo-cycloids. 

It  is  worth  noticing  that  for  both  hypo  -  cycloids  and  hypo  -  trochoids  the  centre  F  is  the  same  for  both  generations, 
and  that  the  radius  f  is  also  constant  for  both  generations  of  a  hypo  -  cycloid,  but  variable  for  those  of  a  hypo-  trochoid. 


DOUBLE    GENERATION    OP    HTPOTROCHOIDS. 

Having  given  the  radii  of  generator  and  director  for  the  construction  of  a  hypo  -  trochoid,  the  method  just  illustrated 
will  always  give  the  lengths  of  the  radii  of  the  second  rolling  and  fixed  circles.  The  accuracy  of  the  values  thus 
obtained  may  be  checked  by  simple  formulae  derived  from  the  same  figure,  as  follows :  — 


Radii  being  given  for  generation  as  a  major  hypotrochoid,  to  find  corresponding  values  for  the  identical  minor  hypo- 
trochoid. 

Let  Ft   denote   the   radius   F  Q     [  =  F  m  ]   of  the   first  director. 
"     Fs         '•         "         "        Fy      [=FM  ]    "      "     second     " 
"     r  "         "         "        R2Q  [  =  Rm]    "      "     first   generator. 

"      p  "         "         "        p2g      [=  pfn    ]     "       "     second       " 

"      tr         "         "      tracing  radius  of  the   first  generation,  i.  e.,  the  distance  R2P2   (or  R  P)  of 
tracing   point  from  centre  of  first  generator. 

Let   t  p  equal  the  second  tracing  radius  =  p  2  P2  =  p  P. 
From  the   similar  triangles   Q  F  q   and   Q  R  2  P2   we  have  F,     :     F,     :  :     tr    :     r 

F.   (tr) 
whence  F2  =     '  v  , (1) 

r 

F,    (tr)                            (  F,              ) 
also  p  =   F2  —   tr  =  -      — -  —  tr  =  tr    \  —   —   1   I (2) 

r  (  r  ) 

and   tp  =  p2P2  =  FRj  =  d,    the   distance  between  the   centers   of  first  generator  and  director (3) 

If  the  radii   be  given   for  a  minor  hypotrochoid  then    FQ     :     p2P,     :  :     Fy     :     p2y, 
from   which   we   have,   as  before, 

radius   of  given    fixed  circle    X    given   tracing  radius 

fixed  radius   desired    =    : (4) 

radius  of  given  generator 

and,   similarly,    formulae   (2)   and    (3)   give   the  radius  of  desired  generator  and  the  corresponding  tracing   radius. 

With  the  tracing  point  on  the  circumference  of  the  generator,  if  we  let  R  =  radius  of  the  latter  for  a  major  hypo- 
cycloid  and  r  correspondingly  for  the  minor  curve,  then 

for   a  major  hypocycloid  R  =  F  —  ?•      (5) 

"     a  minor  "  r    =  F  —  R (6) 

For  the  curves  intermediate  between  the  major  and  minor  hypotrochoids,  viz.,  those  traced  when  the  diameter  of  the 
rolling  circle  is  exactly  half  that  of  the  fixed  circle,  a  separate  division  seems  essential  to  completeness,  and  for  such  I 
suggest  the  general  name  of  medial  hypotrochoids.  For  these  the  formulae  for  double  generation  are  the  same  as  for 
the  "major"  and  "minor"  curves,  and  similarly  derived. 

With  the  tracing  point  ore  the  circumference  of  the  generator  these  curves  reduce  to  straight  lines,  diameters  of  the 
director.  In  all  other  cases  the  medial  hypotrochoids  are  an  interesting  exception  to  what  we  might  naturally  expect, 
being  neither  looped  nor  wavy,  but  ellipses.  The  failure  of  the  terms  "looped"  and  "wavy"  to  apply  to  these  medial 
curves  is  paralleled  by  that  of  the  adjectives  "curtate"  and  "prolate,"  since,  contrary  to  the  signification  of  the  latter 
terms,  any  ellipse  generated  as  a  curtate  curve  is  larger  than  the  largest  prolate  elliptical  hypotrochoid  having  the  same 
director.  And  as  we  have  seen  that,  with  scarcely  an  exception,  "  curtate "  and  "  prolate "  apply  equally  to  the  same 
curve,  our  only  reason  for  retaining  them  is  the  fact  of  their  general  acceptation  as  indicative  of  the  location  of  the 
tracing  point  with  respect  to  the  circumference  of  the  rolling  circle. 

Since  the  medial  hypotrochoids  are  either  straight  lines  or  ellipses,  we  can  readily  find  for  them  that  which  we  have 
found  it  useless  to  attempt  to  construct  for  the  other  trochoidal  curves,  viz.,  simple  terms  suggestive  of  their  form;  in 
fact  the  names  "straight  hypocycloid"  and  "elliptical  hypotrochoid"  have  long  been  familiar  to  us  all,  and  we  have 
but  to  incorporate  them  into  the  nomenclature  we  are  constructing. 

It  only  remains  to  show  that  a  prolate  epi  -  trochoid  can  be  generated  as  a  curtate  peri-  trochoid,  and  vice  versa,  for 
which  the  demonstration  is  analogous  to  that  given  for  the  hypo-curves  and  leads  to  the  following  formulae,  derived 
from  the  similar  triangles  QF<?  and  QRjP,  (the  values  being  supposed  to  be  given  for  the  epi-trochoid  and  desired  for 
the  peri  -  trochoid) : 

=  F,_(*r) _ (7) 

r 

p      =  tr  J  ^  +   1  J      (8) 

tp  =  d  =  distance    between    centres    of   given    generator    and    director  =  F,  +  r (9) 

If  given  as   a  peri- trochoid   and  desired      as   an   epi- trochoid   the   tracing  radius   will  again  equal  the  distance  between   the 


given   centres   (in  this  case,  however  =  R  —  F) ;    the   formula  of   the  radius  of  desired   director  will  be   of  the   same   form 


as  equations   (1)   and   (7) ;    but 

radius   of  second  generator  =  tr    \    1  -      -I (10) 

With    the    tracing    point    on    the    circumference    of   the  generator,   and   letting   K  =  radius    of   the  same  for  a  peritrochoid 

and  r  for  an  epitrochoid,   we  have 

for  the  epicycloid       r  =   R  —  F (11) 

"       "     pericycloid     R=   F  +  r , (12) 


ALPHABETS  AND  ORNAMENTAL  DEVICES  FOR  TITLES. 


ABCDEFG 


&  s  a  L 


No.  1. 

HIJ    KLM    NOPQRS 
I2345678QO 


UVWXY 


No.  2. 

z  L  M 


c  p 


z  3  T 


w  s 


No.  3. 

ABCDEFGHIJKLMNOPQRSTUVW 

1234-5        &        6789O 


XYZ 


No.  4. 

Specimens  of  the  Modified  Italic  Form  called  "Reinhardt  Gothic  " 
its  various  Forms  and  Applications  havinq  been  handsomely  il- 
lustrated by  C.W.  Reinhardt,  in  a  special  Text- Book  devoted  al- 
most exclusively  to  this  Form.  It  is  much  used  on  Engineering 
Drawings,  chief jy  on  Account  of  its  Compactness,  and  its  Legibil- 
ity after  Reduction  by  Photo -Processes.  An  Inclined  Ellipse  is  the 
Basis  of  many  of  the  Lefters.  The  "G"and  "5"  are  peculiar,  al- 
so the  "Q"  Beginners  usually  make  the  Stems  of  the  "p"  b"efc 
too  long.  The  Forms  of  the  Numerals  should  be  particularly 

noted. 

a  bcdefghijk/mn  o  p  q  r  s  t  u  v  w  x  y  z 
A  BC  DEFGH IJ  KLM  NO  PQ  R5TUVWXYZ. 

I  234567890 
abcdefgh  ij  klmn  opq  rstuvwxyz 

A  BCD  EFGH  IJ    K  LM  N  OPQ  RSTU  V  WXYZ.  Half  Section        I          Centering 

of  Masonry.  '          for  Masonry. 

I  234567890 

(The  lettering  above  has  been  kindly  contributed  by  the  inventor  of  the  system,  who,  at  the  request  of  the  author,  has  reproduced  the  remarks 
made  in  the  first  edition,  and  also  presents  the  letters  in  vertical  form,  with  an  illustration  of  the  practical  application  of  his  method.) 


ABCDEF 


No.  5. 

GHIJKLMNOPQRSTUVWXYZ 
12345         A         67890 


No.  6. 


PQRBYUV 

1  1  1 


NO.  r. 


1^34567890 


No.  8. 


,  ,1234567  8  90 


No.  9. 


flDBCDEFQHIJKLnNNOOPQRRS 
TUYWXTZ6"    1    234567890 


No.  10. 


YZ&abedefghijklmnopqrstavuu 
*yz.,  1234567890 


No.  11. 


ABODE  FGHIJ'KLMNOPQRSTUV 

WXYZ&1234567890 


No.  12. 


*  *  »  +  ]  %  3  4  5 

No.  13. 

AHCnEFG-HIJKLMNDPQRST 

UVlATXYZ&ia^   c  d  B  f   gliijklmnn 

12345      ptirstuvwxyz       67890 

No.  14. 

ABCHEF&HIJKLMNDFqHST 
UVWXYZ&aliGiiBfg'liijklinna 


No.  15. 


o 


T 


\z>  o  <d 


No.  16. 


ABCDEFGHIJKLMNOPQRSTUVWXYZ 

I  2345      &    67590 


No.  17. 


i  j  It  "1 


No.  18. 


1234567890 


No.  19. 


No.  20. 


Q  R  S  T  "CT  TT  T77" 

e  f  gr  li.  i  j  3s  1  ZICL 


To  o 


ttt  tt 

1234567800 


No.  22. 


HBCDEFGMIJKLMNOPQRSTUY¥XYZ 

1234567890, 


No.  23. 


T 

op 


T 


cdef 


No.  24. 


No.  25. 


I  JKL 


No.  26. 


8  9 


A        A 
A        A 


n 


"U- 


A     A 


ABCDEFGHIJKLMNOPQRSTUVWXYZ 

&1234567890, 


U  "  V  W  X       :Z 


No.  27. 

1  4^  I<?  CD  AI  0    D  J§  R  & 

TV        1A      AIZ      /  N  v/      isL/      c^Ci      IV     (fj 

Q)        \£>  I 

^Lcdef^bijklmn 

j  ^    y 

2  7  4  r   6yR  Q   o   @>  .  )K  •-• 
J    I  j        /  w 

No.  28. 


abcdefc]hijklrr|nopc|rstuVW*yz4234567o90 


No.  29. 


ABODE  FGHIJKLMNOPQRST 
UV\VXYZ&1234567890 


No.  30. 


(s 


R 


sbedefgfiijK 

2  3  ^'  5 


No.  31. 


No.  32. 


No.  33. 


ABCDErGMUKLnNOPQRS 

TUVWXYZ&1234567890 


No.  34. 


a  fr  t  d  1 1  0  \  \  \  \  1  m 


/ 


n  0  p  qr$t  uu 


No.  35. 


No.  36. 


6  C  D  E  F  Q  tf  IJ  1^  L  M  N  O  P 
Q^STUVWXYZ^-qbcdef 

g  1}  i  j  I1)  1  rq  i]  o  p  q  v  s  \  \\  rf  ^  x  y  & 


No.  37. 


No.  38. 


No.  39. 


X 


abed 


h   i  j   k 


No.  40. 


No.  41. 


Z&C123456789O 


No.  42. 


ABCDEFGHIJKL 
M   NOPQRSTUVW 

XYZ&abcdefghijkl 

m  n  o  p  qrstuvwxyz., 

1234567890 


No.  43. 


ABCDEFGHIJKLMNOP 
QRSTUVWXYZ& 


No.  44. 


e 


c 


56ZZS  9O 


CThe  letters  above  are  the  original  Soennecken  forms,   used  by  permission  of  Messrs.   Keuffel  &  Esser,  New  York,  holders  of  the  American 

copyright  and  agents  for  the  special  pens  and  copy-books  required). 


No.  45. 

ABCDEFGHIJKLMNOPQRS 
TUVWXYZ&1234567890 


No.  46. 


Q  f\  g  ¥  If  V  W  X  Y 


No.  47. 


ABGBEF 
RSI 


*«  1  2  3  4  5 


I  JKLMNOP 
VV XYZ& 
I67590N 


I 


No.  48. 


No.  49. 


No.  50. 


I  IF  (§  LH!  fl  41  K 


ii 


i  i  H 


IP 


No.  51. 


No.  52. 


No.  53. 


H  D  J 


T 


L 

Y 


7 


No.  54. 


No.  55. 


T 


No.  56. 


KBCDEFGHIJK 

L-MNOPQRSTU 

VWXYZ&1234-5 

6  T  5  9  O,. 


No.  57. 


No.  58. 


T        to 

iillabcdBfghijklmnopqr 
stuvwxijz. ,1234367890 


No.  59. 


No.  60. 


t 


^~£~s 


J 


AN  imn*Ha,2UVSS 


WII-*-    II  __      rtK] 

DAY     AND    TO     SLOO     ON 


NOV     f 

OCT  22  1946 


LD21-50m-8,'32 


01975 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


